When wires get this large, using copper is just throwing money away. 2AWG aluminum is a very appropriate choice for this application. Whatever ooga-booga you may have heard about aluminum, never applied to the large conductors like this, and doesn't apply to the new AA-8000 alloys.
How about EMT steel conduit? Use that, and you don't need to pull a ground wire. The EMT is the ground. (Not only that, it's ground for any other circuit that needs to be grounded back to that panel, in case you are retrofitting grounds in an older home).
If possible, I would use 2" conduit. I know the tables say 1.5" is adequate, but this assumes you have an electrician's truck chock full of pulling tools and I'm guessing you don't. The more oversize you go, the easier the pull, and the less likely you'll need to hire the guy with the truck. And guess what, he won't work with you unless you hire him for the whole job. I managed to hire one just for the pull, but he was a friend of my family, and even he needed to delay the job a day so he could buy another tool he didn't have. Anything you can do to ease the pull is a good idea.
Watch the bends. Every bend makes the pull much harder. When I lay conduit, I allow myself only one 90 degree bend between accessible points (Code allows 4, but you'd need a power winch and a bunch of lube to pull 3x#2AL around 4 of them! Yikes!) Whenever you use a junction box or conduit body at a 90 degree bend, that replaces the bend with an accessible point! Quite a bargain! It also makes the turn sharper and neater. If it were me, on a big one like this, I'd put an access point at every single 90... which would make the pull easy, and could make it workable with smaller pipe. I still wouldn't go smaller than 1-1/2".
Food for thought (i.e. where to use a junction box instead of a conduit body), you can have up to 4 circuits in a single conduit. (literally, 9 conductors, but grounds don't count, and neither do neutrals on split-phase 120/240 circuits and MWBCs, so you've only used 2 of 9.)
I quite agree, pulling three 4/0 Al wires is the kind of job a DIYer ends up getting stuck on and having to call an electrician just for his truck full of pulling tools.
How old is this work? If it was legal to wire ungrounded at the time, it may be legal to retrofit ground using the NEC rules allowed for that. For a retrofit, the rules are liberalized and you have a much wider set of choices for how to get it done. Make that argument to the AHJ and see what they say.
Also, why dig a wnole 'nother trench? You only need to dig out the old trench to the 45, which I assume is only a few feet from the house and shed. Then you can disconnect the wires, cut the pipe (carefully) and get the access you need to get the ground wire through. If the joints were not cemented, it'd be even easier. Really keeping an underground conduit watertight is a lost cause, it will fill up with water, that's why the wire has a THWN rating. So not cementing it does make sense. It's exactly what I'd do in that circumstance.
Also, if your wire is the obsolete AA-1350 type, just be careful with your terminations, make sure the lugs are made of aluminum (most are), use the anti-ox and torque them exactly to spec. I see no reason to spend a fortune on new AA-8000 wire, they both conduct electricity just fine, the issue was always at the terminations, and even that was mainly an issue for the tiny wires (10-12 AWG) on Cu terminations. On the other hand if the old wire is not outdoor rated, that's a different kettle of fish.
Best Answer
A brief visit to the code indicates that there is a bigger problem, in that the ampacity of 8Ga copper is only 55 at 90C (aluminum 45 Amps) so your wire is too small for a 100 amp service. You need a bare minimum (if everything is rated for 90C) of 3Ga copper or 2Ga aluminum, and probably larger after various derating factors are applied, or if 75C is the limit on any connection. There is a note that may permit use of 4Ga copper for 100 amp in a dwelling unit as a "service lateral" - you'll have to decide how you feel about taking that note seriously. I've linked to a handy calculator that might be instructive.
Ampacity calculator for branch circuits and feeders
Non-pickily, for a wire appropriate to the ampacity of the subpanel, no. If you are using an 8-3 w/g cable, the cable is probably fine, and being a cable would complicate running it in conduit (as opposed to 4 individual wires.)
Pickily, maybe, if the wire is running through an area where it's prone to damage, or through a floor - but in both those cases, the conduit is only needed in the affected area, and is a "sleeve" rather than a "raceway" for code purposes.
If you don't already have the 8-3 w/g cable bought, do compare costs .vs. THHN or THWN (or XHHW, etc) in conduit. Edit: Well, even if you do have it bought, you can't feed 100 amps with it, so shop around for bigger wire.
I have a personal bias in favor of conduit, but that's all it is. I've seen too many scary things in remodeling where rodents have stripped NM cables in concealed spaces. In addition, you are limited to the 60C rating when using NM cable.