Gunni Omann Credenza Needs New Doors! Veneer??  

Try to find an antique furniture restorer that been in business for a long time. They often have a good variety of veneers on hand or know where to get unusual pieces. Museums can sometimes provide good referrals.
Ouch, though. Those door panels are gonna be hard to repair well. And right at eye-level, too.
Making "dutchman" repairs in the middle of a field like that is almost more of an art than anything and there are restorers that can make a presentable job of the seemingly impossible... if you can find them. I suppose re-veneering all of the doors with book-matched leaves of veneer is perhaps a less challenging option, though not without its own set of drawbacks and challenges. I guess I'd recommend making a lot of inquiries and taking your time weighing pros and cons as they apply to your goals for the piece before jumping in. Good luck!

What do the backs of the doors look like?

They are splintered out, one is really bad, from where the object hit the door.

How about getting the depressions filled and then painted to match the grain? I have seen this (in person, not just photos online) done so perfectly that you have to get within inches to tell that it isn't the original wood.
You will have to hunt around to find someone with these skills but it seems to me like a much better solution than trying to find veneer that will match the rest of the piece, or that will have to be dyed or stained to match the rest and then who knows if the new color will be stable over time.
The fortunate part is that the doors can be shipped easily, so you don't have to limit your search to local people.

I would seriously look at the fill and faux paint option. Some people really are amazing at this.
As far as a source for veneer, I would go to ebay. One of the very challenging problems you are going to have is that teak quite substantially changes it's appearance over the time with exposure to sunlight. Unexposed teak is much more brown, and shot through with black streaks. Exposed teak looses the brown, and the black streaks diminish almost entirely, leaving you with the orangish wood we are familiar with. So it is going to be extremely challenging to find new veneer that matches.
Is the piece finished on the back?

I love the idea of painting it! Didn't think of that!! Do you guys know of any businesses off the top of your head that do this? I was googling and found a little, but nothing that I've been sold by. Thanks so much for your suggestions!