Can an Officer who is a 91A switch to 89E? What's necessary?

Spoiler up front: LTs (2LTs/1LTs) are not normally eligible for VTIP.

If you're active duty, you'll want to read MILPER 22-422* which has the details of this FYs Voluntary Transfer Incentive Program (VTIP) and the HRC's webpage on the program*. ARNG follow the procedures of Chapter 6 in NGR 600-100 and USAR are covered in Chapter 6 of AR 140-10.

You need to look at the VTIP IN/OUT chart* to see if your branch is releasing you or if the other branch is accepting you. If neither applies, you can always contact HRC (instructions on website) an ask for an exception (worth a shot).
---------------------------------------------
* MILPER Message 22-422 - https://www.hrc.army.mil/milper/22-422
* HRC VTIP Website - https://www.hrc.army.mil/officer/officers%20voluntary%20transfer%20incentive%20program%20VTIP
* VTIP In/Out Chart - https://www.hrc.army.mil/asset/21170
* AR 140-10 - https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN30735-AR_140-10-000-WEB-1.pdf
* NGR 600-100 - https://www.ngbpmc.ng.mil/Portals/27/Publications/ngr/NGR%20600-100_20200706.pdf