It's possible to cross the Uganda-Kenya border in the caldera of Mount Elgon. However, there's a lot of missing information on the subject and the Ugandan Wildlife Authority (UWA) provided essentially no information when asked.
I looked for a blog post on the subject, but couldn't find anything. So, here's the post I would have liked to find online! This is current as of April 2018.
What you need to know before making calls
You will need to choose which trail to use to ascend the mountain from the Uganda side. Sasa (cross on the third day) and Sipi (cross on the fourth day) are the two most popular. Sipi has you camp in a giant cave of bats; Sasa has a cool section of staircases up a cliff (they are referred to as "ladders", but they are definitely stairs).
It is up to you to know at what time the border crossing "handover" will occur. Your best bet is to calculate backwards: if you want to get your passport stamped on the day you cross, you should estimate 3-5 hours hiking on the Kenyan side (9 kilometers with a few step hills and no particular trail before the descent). After that is a ride in a 4WD military vehicle down to the KWS (Kenya Wildlife Service) park office– about an hour, and maybe another hour of paperwork. Add another two hours for transit to the border and back, which they can arrange from the office.
The border closes at 5pm, so if you make the handover after 9am, you may already be too late.
If you don't mind a night in the park on the Kenya side ($40 for a 3-person banda with kitchenette and theoretical hot showers), a handover time of noon is fine.
Call the Kenyan side before talking to the Ugandan side so that you can give the UWA official the phone number of the person who is arranging your Kenyan guide. You want your Ugandan guide and your Kenyan guide to be in phone contact. Make sure this contact occurs before you hike out of phone service on the first day.
Call the KWS and let them know your plans. They will give you the phone number for someone who can arrange guides and porters for you.
Unlike the Ugandan side, the guide is not included in the park fee. You don't have to have one per person. But you will be met by "security"– I'm not sure…