The owl would in no way benefit from Improved Unarmed Strike

Strictly speaking, this isn’t true—a giant owl can make an unarmed strike as appropriate for a creature of its size. It just ordinarily wouldn’t have any desire to do so.

Anyway, no, there is no rule that waives the Improved Unarmed Strike requirement on Improved Grapple for creatures with natural weapons. Most creatures that are intended to be grappling get it as a bonus feat, or get improved grab, or both.

There are various ways to get Improved Grapple without Improved Unarmed Strike if you take the right class levels—bear-totem barbarian 2nd is probably easiest, though monk 1st is even easier and then you just pick up Improved Unarmed Strike for “free”—but animal companions cannot get class levels. Strictly speaking, I guess, a giant owl could probably get the leg of squid graft from Dragon vol. 318, but at 40,000 gp, that’s pretty rough.

The real answer to this is, if you’re “designing an encounter” as a DM, is to just give the giant owl this feat. Remember,

You can add any sort of spell-like, supernatural, or extraordinary ability to a creature.

^{(Improving Monsters → Adding Special Abilities)}

“A giant owl can ignore the Improved Unarmed Strike prerequisite on Improved Grapple” could be an extraordinary ability of approximately zero value.

The owl would in no way benefit from Improved Unarmed Strike

Strictly speaking, this isn’t true—a giant owl can make an unarmed strike as appropriate for a creature of its size. It just ordinarily wouldn’t have any desire to do so.

Anyway, no, there is no rule that waives the Improved Unarmed Strike requirement on Improved Grapple for creatures with natural weapons. (Was incorrect on this point: see HeyICanChan’s answer.) Most creatures that are intended to be grappling get it as a bonus feat, or get improved grab, or both.

There are various ways to get Improved Grapple without Improved Unarmed Strike if you take the right class levels—bear-totem barbarian 2nd is probably easiest, though monk 1st is even easier and then you just pick up Improved Unarmed Strike for “free”—but animal companions cannot get class levels. Strictly speaking, I guess, a giant owl could probably get the leg of squid graft from Dragon vol. 318, but at 40,000 gp, that’s pretty rough.

The real answer to this is, if you’re “designing an encounter” as a DM, is to just give the giant owl this feat. Remember,

You can add any sort of spell-like, supernatural, or extraordinary ability to a creature.

^{(Improving Monsters → Adding Special Abilities)}

“A giant owl can ignore the Improved Unarmed Strike prerequisite on Improved Grapple” could be an extraordinary ability of approximately zero value.