Generally1, the Polymorphed creature can't grapple
My reading on the other action is other than spells. Grapple requires a free hand, thus it obviously requires hands.
The target of your grapple must be no more than one s/ize larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an Attack roll
This reading means polymorphed creatures can not take the Grapple action. (Yes, I'm aware Grapple is not an action - it's just a special Attack action, but I think this way is easier to understand.)
There are exceptions
Monsters don't (necessarily) follow usual Grapple rules, as stated in the MM p. 11.
Many monsters have special2 attacks that allow them to quickly grapple prey. When a monster hits with such an attack, it doesn't need to make an additional ability check to determine whether the grapple succeeds, unless the attack says otherwise. A creature grappled by the monster can use its action to try to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check against the escape DC in the monster's stat block. If no escape DC is given, assume the DC is 10 +the monster's Strength (Athletics) modifier.
This leads to some monsters being able to grapple creatures without actually requiring hands. For example, a Giant Octupus has the action
Tentacles. Melee Weapon Attack: +5 to hit, reach 15 ft., one target. Hit: 10 (2d6 + 3) bludgeoning damage. If the target is a creature, it is grappled (escape DC 16).
This action does not require hands, thus it is allowed. The action states that the creature is grappled, so the creature is grappled. Thus, Giant Octopus can grapple normally through the use of Tentacles.
For other examples, T-rex and Giant Toad also have actions that grapple using Bite, which is more far away from "hands" than the tentacles mentioned above.
Crawford has also spoken on these matters:
Q: Polymorph says I cant take actions that use hands. Grapple action requires a free hand. Can Polymorphed PC use grapple?
A: I would allow it if the beast form had an appendage or a bite that could potentially restrain someone.
Note that this doesn't follow the usual "These are the official rules" style of writing from Crawford - it just says "I would allow it", making unclear if that's how it actually works. This is an old tweet, though, and that might be the reason for it. Also there is a lack of "allow only if", but I guess that's implicit easily.
Possible house rules
Personally, I don't see any (balance/mechanical) problems with allowing a Giant Ape to grapple normally. I would think the restriction of using hands was more for locks, doors and other things requiring Dex than actual combat actions. My reading on RAW and Crawford's tweets say this isn't the official rule, though.
1 Here, generally means that it's the rule that holds true unless stated otherwise, e.g., unless the beast has a specific action that lets it grapple without hands, as described in the following section. The fact that this is the general rule is the reason beasts are required to have the specific action to be able to grapple as we would usually imagine, for example, a Boa Constrictor doing.
2 Since the "general vs specific" discussion is constant, I'd like to emphasize that the Monster Manual itself says these attacks are special, i.e., not the general/usual rule. These are exceptional, special, specific attacks described in their statblocks that apply exclusively to the monster in that statblock, thus it can't be general in any way. Even if many monsters have these special attacks, all of them are specific to each monster, and the general rule is still applied to all of them.