Yes, they can stack, though it is tricky to actually use in combat.
Note the difference between the description of Deadeye Shot feat and the actual rules text listed in the benefit:
When your target is busy dealing with an ally's melee attack, you strike
Actual Rules Text (Benefit):
If you ready a ranged attack to occur when an ally strikes a particular target, and your ally succeeds, that target loses his Dexterity bonus to AC against your attack.
The rules text does not restrict what kind of attack your chosen target is distracted by, even though the description assumes melee. There are several steps here:
- pick a target.
- ready a ranged attack.
- any ally successfully strikes your chosen target.
- that target loses Dex to AC versus your ranged attack.
This does in fact combine with Manyshot allowing sneak attack - or whatever other ranged precision damage your character may have which would trigger due to a target losing Dex to AC - to affect them via your ranged attack.
However, in Manyshot, only the first arrow qualifies for precision damage. The other arrows do not benefit from it.
Consider then the Greater Manyshot feat (found in the Expanded Psionics Handbook and also the SRD, psionic feats section, for those who are restricted to core only) which allows one to attack one or many targets as the character chooses, and also roll an attack roll for each attack, applying precision damage to each arrow.
Note that the attack penalties from Manyshot still apply, so keep that in mind when picking your targets. Unless you have an insanely high To-hit bonus, perhaps save this tactic for use against mobs of mooks with low AC, instead of boss-types that typically have high AC.
Note also, that not only would you have to ready an action (which is a standard action), but you would have to wait until the next round before you could attack with it, as you've already used your standard action for the round by readying the action and Manyshot (or Greater Manyshot) is also a standard action.
There are ways around this, such as items or powers which can grant you extra standard actions in a round, but unless one is willing to tolerate a fairly high level of optimization and rules-min-maxing, this would be a difficult and/or expensive tactic to sustain other than as an every other round attack.
To be honest, you'd probably be better off ditching Deadeye Shot and having someone use Grease on your targets combined with using Greater Manyshot every round. (The trick with Grease is that anyone who has to make a balance check automatically loses Dex to AC even if they succeed on the check. There are some specifics and exceptions to this, so check the Balance skill for details.)