Yes, animate objects can allow instruments to play themselves
Animate objects (PHB, p. 213) says:
[...] Each target animates and becomes a creature under your control until the spell ends or until reduced to 0 hit points.
[...] you can mentally command any creature you made with this spell if the creature is within 500 feet of you (if you control multiple creatures, you can command any or all of them at the same time, issuing the same command to each one). You decide what action the creature will take and where it will move during its next turn, or you can issue a general command, such as to guard a particular chamber or corridor.
From the above, it seems clear to me that you can command all of the instruments to "play", which given that they are instruments, they should be able to do without issue. That said, if, say, you commanded them to play themselves, then their Charisma (Performance) apparently wouldn't be very good, because:
An animated object is a construct with AC, hit points, attacks, Strength, and Dexterity determined by its size. Its Constitution is 10 and its Intelligence and Wisdom are 3, and its Charisma is 1.
That said, as DM, I would simply allow them to be played competently because they are musical instruments, but other DMs may rule differently (perhaps if they ruled that the instruments lacked any creativity to perform well or something along those lines).
Generally speaking, though, from a more narrative perspective, there are only two reasons I can foresee where this would happen; either you're the DM and you want a cool scene where instruments play themselves, in which case of course it should work, or your are a player who wants to do something cool with one of your spells, in which case I would encourage any DM to allow it to work and create a cool moment.