Supportee is a word. The Ngram viewer finds it's used particularly for emotional support:
Gottlieb explored several different support situations and assessed the degree to which various support behaviors are considered helpful to the supportee. (On Being Supportive: The Emotional Consequences of Listening to Another's Distress by R M Perrine)
and for physical support:
Support-edges for support relationships in the Initial State are called disassembly-support-edges. They are shown as solid, u-shaped edges. These edges are directed from the supportee to the supporter. (Intelligent robotics: proceedings of the International Symposium on Intelligent Robotics, January 2-5, 1991)
and in the discussion of tax law:
In the case of a charitable trust...an alternative responsiveness test could be met when the supportee was named, and the named supportee had the power to enforce the trust.... (Tax Planning and Compliance for Tax-Exempt Organizations by J Blazek)
Checking with the google finds that supportee has been used for product support. From the website of a desktop conferencing service:
You can use GoMeetNow to remotely support PC and Mac computers. Follow
the procedure below:
- Start an interactive meeting session & invite the supportee
(attendee) to join your session
- After supportee joins your session, change the presenter to
supportee.
- After supportee accepts to show his desktop, ask him to assign controller to you.