[RPG] Can you buy the unusual items provided by backgrounds

Part of the problem is that the standard conception of an adventurer is somewhat unrealistic in and of itself. An independent group of characters who possesses a great deal of material wealth and specialized equipment yet travels so widely that they seldom cross the same patch of ground twice is extremely rare in realistic settings.

With that said, there are a few things you can do to make it more workable:

Be prepared

The biggest thing that can help you out here is to get information ahead of time about the adventure you're going on. Figure out what you need, and what you can leave behind.

Strip down to the essentials

Look at Strider in the Lord of the Rings movies as an example. His gear is extremely streamlined -- He carries no tent, no utensils, few rations, little in the way of personal effects. He uses his survival skills and a bare minimum of equipment (some means of starting fires, a cloak in place of a blanket, and so on).

Naturally, you'll need some form of survival skill to make this work. And the specific environment you're in will determine just how much you can throw away.

Quick release packs

For small loads, you can make it a practice to drop your packs when combat begins, and then pick them up again when the immediate danger has passed. This still leaves you somewhat vulnerable to surprise (and traps) but otherwise allow you to function "above load."

Caches

Bury your excess stuff at critical locations. If you're in the process of bringing a haul back to town, bury some of it at the dungeon and make several trips. If you're about to leave town for the adventure, bury any gear/food/etc. you won't need. Make a habit of creating survival packs and hiding them in areas you frequent.

This was a tactic employed by real-world people who had more stuff than they could carry (I most recently read it in relation to people taking loads of gear up into gold country during the northern gold rush).

Use magic

Bags of holding were invented for a reason.

You have 2 choices for starting equipment. One is to take the equipment given by your class and background, which in some cases (such as the Rogue) includes tools, and in other cases (such as the Fighter) does not.

The other option is to take starting gold as given at the start of Chapter 5, and buy items from the basic equipment lists up to that amount. If you want a tool that your class and background do not provide, you can take starting gold and buy that tool. (Note that if you choose to take starting gold instead, you do not get any equipment from your class or background.)

You should consider, though, that in general the beginning of your life as an adventurer is meant to be a significant change from whatever you were before. (Such as a criminal, or urchin, and so on.) So even if you had those tools in your past, and became proficient with them during your former occupation, whatever caused you to become an adventurer might have involved leaving some possessions behind, or some things been stolen from you.

It's not so much a case of "proficient with tools she does not own" as "proficient with tools she had at some point but doesn't have now for some reason." But as I said earlier, if you want to have something that your starting equipment does not provide, you're free to take starting gold and go shopping for it.