I'm not familiar with the community nor RDR2 speedrunning, so my opinion might be nonsense
It looks like R* has a website ( http://wheelerrawson.com/ ) where you can purchase items for the in-game characters, and it works for both story and online. (Purchases for online are sent to post office as if purchased through catalog, and happens instantly. Purchases for story will be directly added to your inventory, and happens when you re-open the game.)
GTA5 has a similar service (a smartphone app called iFruit that allows modification of in-game cars), and the GTA5 community decided to ban the use of this service (technically they banned everything that requires "an active internet connection) in case R* shuts down the service in the future and cause future runners to potentially have a disadvantage.
In the current RDR2 rules, I don't see a clause explicitly banning the use of online services - should they be banned?
edit: in case anyone wants to argue "it doesn't give advantage in runs anyway", GTA5's iFruit also doesn't give a speedrun advantage, but they banned it regardless.
There is no rule for or against it because currently you have to restart the game for these items to be in game. I don't think it warrants a ban as of right now because I don't see it having a huge or any advantage
Re Demomute: I think I saw something that has something to do with a loadless timer. If the time taken to restart the game is removed by a loadless timer, then one might use the web shop and restart the game with little time added to the loadless timer. Again, I'm not familiar with the speedrun route, and I'm not sure if there is any unscripted shopping needs that might take advantage of the web shop.
Re FreshRigi: the point to ban web shop is to future proof. Once the web shop eventually closes down, any runners that run the game after that point will have a disadvantage on the leaderboard.
ah yeah, future proofing could be a problem. it really does not matter too much, though. 100% is the only category this is useful for and over the course of that 30 hour run it doesn't make a significant difference. i see no reason to ban it.
The website allows buying guns way before they are actually unlocked. On a save just after A Strange Kindness, I am able to buy Mausers, the Carcano and the Repeating Shotgun.