Yes, it would not necessarily be very good, and it would likely not be digital, but there is no reason that such a computer could not be built. Animals may not be very reliable, and they may drop dead at unfortunate moments, but so do computers - admittedly, the more animals you have the less reliable you can expect things to be, overall, but something does not have to be very reliable to be useful - think of early weather reporting - nor does it have to be reliable at all, to be considered by a lot of people to be useful (think of horoscopes), sometimes just the novelty (the newest new thing) is enough to get people either using it, or enthusing about it.
As I said, it would likely not be digital, but could be very good at fuzzy thinking and pattern matching. A lot of modern computing is actually based around recognising significant patterns and taking actions accordingly. The trick is to reduce problems to patterns to be matched. Patterns could probably be best matched across the workforce, who of course would all need to be fed (think Bites, rather than Bytes), although some motivation other than food would be needed to drive the actual processing. It needs to be carefully thought through, but I believe it would be possible.
The main obstacle, as I see it, would be designing a method of data storage and retrieval - I have no idea how that could be done, but I expect some imaginative person could think of a way, or a method of obviating the necessity.
I would suggest using small animals, such as ants, rather than larger animals, like mice. I would also suggest reading Douglas Adams, 'The Hitchhikers Guide to the Galaxy'.