My group uses a private Google+ community to organise the games, and G+ alongside Roll20 to play the games. We play entirely online on a weekly basis, mainly because we're all about the same age and have small children and families, so it's difficult to get together for a face to face game.
For D&D 5th Ed you're pretty spoiled for choice, but when it all boils down the best two options are this:
Roll20, either in conjunction with G+ or on it's own. You can create an account and run your campaign through it for free, and the biggest pro (in my opinion) was the ease of use when it integrates with G+. Most everyone has a google account these days, so they also have a G+ account whether they know it or not. Once you've created your game in Roll20, you can choose to launch it in a g+ hangout.
Once you've done that for the first time, whenever you launch a G+ hangout the Roll20 app will be present, and you can launch it and select your campaign from within the hangout. Running the game each week becomes as simple as scheduling a G+ hangout event and showing up - the players don't even need Roll20 accounts, they just click the hangout invite link and game on. The D&D 5e character sheet for Roll20 seems pretty excellent too, with some dice rolling automation built into it. We're taking it for a test run soon hopefully.
The other main option right now, though it can get pricy, is Fantasy Grounds. It's not as simple for the players as Roll20 (which is, as i've said, just a link click), but if they're willing to install the program (available on Steam now as well), Fantasy Grounds is now the official online tool for D&D 5th, in partnership with Wizards of the Coast. It has excellent integration with the rules system and character sheets, as well as supporting hero lab. Some friends of mine swear by it, though it is a bit more complicated to get set up and running.
Both solutions support full voice, and I know Roll20 supports full video both in the app itself and through hangouts, as well as an online tabletop for maps, images, handouts and whatever else you like. This made it a lot easier for us to retain that "social tabletop" feeling, despite playing entirely remote. Using a large enough screen and a good set of speakers and microphone, you could play partial remote (with only a few players remote) or even with everyone in the room and disabling the voice and video altogether if you wanted.
Either way, if you do plan on getting in to running your games online through virtual tabletops, Google+ is your friend! There is a massive community of people there who will be more than happy to lend any assistance required.
Personally, I run a community geared at assisting people to set up, run or find games in the asia-pacific time zones. You can find it here: https://plus.google.com/u/0/communities/100566740249118526694
For other timezones, such as the American zones, a great place to start is here: https://plus.google.com/u/0/communities/107422815813208456454 It's an excellent community filled with friendly people.
You can run an AL game no matter what anyone else says.
From the Adventurers League website:
It is simple and easy to become a Dungeon Master for the D&D Adventurers League. You do not need to register or log your play anywhere online. Simply ensure you are running games that follow the D&D Fifth Edition rules and the Adventurers League rules, and ensure your players are using characters created with the Adventurers League rules.
You, as described, are doing all these things and so are perfectly within your rights to use the AL trademark to describe your games.
When DMing in a store ...
Now the store may only be happy to host you if you participate in whatever they've set up, including their delegation of some authority to a store "coordinator." I'm not quite clear on whether that's happening: you say the employees don't know anything about AL, but you also say there's a store site where AL is posted and apparently the store "supports" some AL games. There may be more going on here that you aren't aware of?
Back to Adventurers League....
AL used to be, well, a League. There were site coordinators and regional coordinators and you had to sign up and have a password and whatnot. (I was a site coordinator for one site and a GM at another.) Circa season 4, with the launch of DMs Guild, all that went away. Adventurers League is, frankly, now a set of published play-conventions.
This is all to say that this "self-appointed organizer" has no business telling you what to do in order to bill your game as AL. But they may be used to being in the position of telling other AL DMs what to do, if they were a coordinator.
Solving your problem
You need to get the store involved, in my opinion. You are trying to run a game in the space they provide (presumably hewing to whatever rules the store has for that). Another store patron is harassing you. (Again: it may be the case that they're coming from a place of confusion or habit, not malice.)
There are "official" organizers of AL who these days mostly (to my understanding) work with conventions and the online community. You can find their contact info either at the website or on the AL facebook page (very active, that one), but I don't think that's the best way to go. Having been a site coordinator for years and then having that support disappear four years ago, I advise just working with your store.
Best Answer
Here are a few things I ususally pack for in person gaming:
Adventure Book. Don't forget to bring it. It has happened to me before.
Dice. Obviously. Bring enough multiples of the types you need to speed things up, especially d6 and d20 for rolling things like falling damage, fireballs, or attacks by multiple monsters.
Master Screen. If you want to hide your notes, dice rolling, and the open adventure, and to look up common reference materials a Master Screen is a great tool. It is not a must have, but quite helpful. You can also have a sheet there for notes, timekeeping and tracking monster hp in fights.
Rule Books and page markers. To look up things, you can use your mobile phone or ask players to use theirs, but the rule books, in particular the Monster Manual, are also useful. Best have some markers, Post-Its or bookmarks to stick in so you can quickly find what you need.
Floor Plan We usually do not print the plans, we draw them on a coated gridded plan, which is fast and very flexible. You obviously can also preprint them, and try to cover them, but in my experience, that is more cumbersome than just drawing what people see as you go along. You also can draw on a sheet of graph paper, and only move to the big plan when combat happens.
Tokens or Miniatures. Printed pogs can work well, having 3-D minis makes for a nicer visualisation. We find the clear plastic boxes in which some dice sets come quite useful to indicate flying creatures, too (you can put them over other pogs/minatiures).
Pencils, Paper, Boardmakers You may want to write or draw on the map and make notes. If you use pencils, make sure you also have a rubber. For markers, best are ereasable markers that you can wipe off again, if you have a coated battlemap.
Initiative Tracker There are many ways to do this. But you need one.
On site you will also need:
Space. Make sure you have a large enough table for the battlemap, best with space for the character sheets, drinks, miniatures, rulebooks, dice rolling area to spare.
Optional: Food & Drinks. You are playing in person, so you can share food. Make sure you have enough, and depending how long you play, maybe also have some healthy option, not only sweets and chips.
Optional: Power Outlets, especially if you are playing longer, and there are multiple people that want to charge their phone or laptop. We sometimes bring a power strip.
Optional: Music. We often have thematic music running in the background to enhance the atmosphere. It can become a bit of a distraction if you fudge around too much with it, but there are also soundscapes like in the inn, forest sounds, battle sounds. Our DM usually brings a compact speaker and streams it.