Yes, they most definitely do AI in miniatures. And no, you don't need to get on your knees to collect them <smile>. We have done quite a bit of semen freezing of miniature stallions for Little King Farm, including Boones Little Buckaroo. Additionally, yes, there are AV's that are designed for the <cough> less well endowed of the equine world ;).
The registries do allow the use of AI in miniatures, as well.