Anyone know if the Medical CoE is experiencing any problems with scheduling AIT slots?

PFC (Join to see) her unit readiness NCO SHOULD know what is going on, if not he can contact others that WILL know what is going on and what the projection of training seats are for the SM. Now whether or not that information is something the readiness is tracking, is motivated to find out..that's a different story. It may take a conversation with the SM's chain of command to motivate the readiness NCO.
And Id suggests that conversation is done in a positive, professional request sort of way.
"Sir(ma'am), I'm very much looking forward to completing AIT, rejoining our unit, and being able to more completely support the mission. I'm also doing my best to make prior plans for attending my AIT so there are no or fewer last-minute arrangements that could make me miss or delay my training. Sir (ma'am), could you look into when my AIT training date is scheduled, or if none, what if any anticipated date ranges can be expected?"

All depends on training seats, I’m sure her Readiness NCO is tracking already. Some MOS’s are only offered in certain times of the year.