Web, as well as most other "hazard" type spells, do not trigger the Gravity Well feature.
Outside of the question of if casting Web to include a creature in its area counts as "casting it on a creature", the largest reason why I say this is because Gravity Well specifies "when you cast". This means we can't apply this on later turns, we can only apply it at the time the spell is being cast.
Since Web or other "hazard" spells like Flaming Sphere or Moonbeam don't apply an immediate saving throw on cast (as detailed in this answer), only at a later point in time, Gravity well never has a chance to trigger. Grease, on the other hand, would work with Gravity well, since it applies a saving throw right away, but only when you cast the spell, as detailed above.
You further state that you would like to use this spell to move allies out of its area of effect when you cast it, if possible. Unfortunately, that won't be possible either.
First and foremost, Gravity well requires that you cast a spell "on a creature" (it should say "targets a creature", but it doesn't, and so, it technically doesn't follow the rules for targeting creatures). Casting a spell "on a creature" means that as part of casting the spell, you specifically select a creature to affect as art of casting the spell, such as casting Hex, Booming Blade, or Chain Lightning. In these instances, you have "cast a spell on a creature". Each requires that you select a specific creature (or multiple specific creatures) to be affected. Web does not do this, instead, you cast the spell on an area. Since you've cast it on an area, instead of on a creature, you can't move anyone inside the web, even if they are allies who are willing to move.
Now, in my comments, I've also proposed changing the text of the feature from "cast a spell on a creature" to "cast a spell that targets a creature". This will expand how the ability functions, but it still won't work with Web. This is because of how targeting works. In 5e, any creature directly affected by a spell is a target. If you're inside the area of Fireball and have to make a Dex save, you're a target. If I cast Hex on you, you're a target.
If you cast Web, anyone who starts their turn in the area, or when someone enters it for the first time on a turn, they become targets for the spell, but they are not targets prior to that moment. This means that Web can never be used to move creatures, because it doesn't affect them when it's cast, it only can affect them after the casting has finished.