Primate Dexterity will lead to overpowered "tanks" (and maybe overpowered high level casters)
Many builds rely on the ability to stay in one place and either attract or lock down other enemies. To these characters, mobility is rarely an issue. The main choice they have to make is between damage output and defense: in practical terms, whether or not to use a shield. A shield adds a precious +2 to AC, but deprives them of a free hand stopping them from using either Two Weapon Fighting, or Two Handed Weapons.
This race would enable you to stop making this choice and have it both ways. They could wield a shield in one hand and use a hand and foot to use a two handed weapon, for example: they could even take a foot of the weapon (once they've made all their attacks) and move around as much as they wanted. If this was considered "dropping" the weapon, they could even put their foot back on the weapon at the end of the turn (since dropping an item typically doesn't take your "interact with an object"). This racial trait therefore basically adds a +2 AC to such characters (or higher if they have a magical shield), which is a major benefit in a bounded accuracy system.
It also could be used by certain "blaster caster" types. There are magical items which give bonuses to spell attack rolls made "while holding" them. Such items include the Wand of the War Mage, The Staff of Power, and the Rod of the Pact Keeper. The game is currently balanced around the assumption that a spellcaster could benefit form at most two of these magical items at a time. The ability to benefit from three could unbalance a higher level character.
Generally, trade like for like
If you want to keep things balanced, I'd suggest two major changes. First, I'd suggest that magical items held in feet (other than footwear) do not give any benefit (reason that they are designed to be used by hands). Second, I'd recommend that you require an Action to grip an item with a foot, and maybe another Action to release it (reason being that the feet are very dexterous, but they instinctively grip items held in their feet as hard as they can). This means that every time they shift from using two limbs it has an "opportunity cost" that prevents them from abusing this extra limb for cost-free extra attacks.
Take the Tabaxi as a guidance for designing racial features that add serious benefits in combat. The Tabaxi have the feline agility feature, which states:
Feline Agility. Your reflexes and agility allow you to move with a burst of speed. When you move on your turn in combat, you can double your speed until the end of the turn. Once you use this trait, you can't use it again until you move O feet on one of your turns.
This is a massively useful ability, as doubling your speed massively increases mobility (and combines well with other features, such as the spell Haste). However, in order to use this feature again they have to forgo any movement in another turn. This means that their average movement speed over many turns will be similar to those of another character (who moves 30 feet a round, rather than 60 feet every other round), keeping things balanced.
Essentially, the tabaxi gains movement at the cost of later movement. That is a fair and balanced trade. In your case, you've essentially traded damage and/or AC for movement (and maybe not even that, depending on how the characters use this trait). This isn't a fair trade, and will likely result in an unbalanced system.
There are components of racial benefits that give unambiguous benefits to damage. For example, a Lizardfolk can do 1d6+Modifier damage with an unarmed strike (bite), which is an average +1 damage per attack for a low level monk. However, even this is a tradeoff for damage, because Lizardfolk don't get bonuses to dexterity, most Monks' main attack stat. When designing a race, you have to think of whether or not this will make an unambiguously better version of any given class than every existing race. And in the long run (excluding the first few levels where Variant Human will have a clear advantage) this will make a Fighter with a pretty clear advantage over all the other Fighters out there.