The Illinois Condominium Property Act requires that the board be composed of "unit owners" and the definition of "unit owners" is the owner of "fee simple absolute" title. Where title to real estate is held in trust, the trustee normally holds fee simple title, which is "record" title. You want wait for the HOA to elaborate but unless there is something else going on, those are the basics.
The best answer to your question will lie within the rules and regulations of your Homeowner's Association. Many HOA's maintain a website where you can download them. If not, contact your HOA and ask them for a copy of the HOA rules and regulations,
I am not licensed in Florida and as you have asked this question out of McHenry, Illinois, you are likely to get only Illinois attorney answers. There may be some HOA prohibitions that are Florida specific and as such, it could not hurt you to re-post your question using the city or area in Florida of where the property is.
In general, I would think that if you live in the home that falls into the purview of the HOA, I would have to initially challenge a rationale that having a home in a trust would exclude a party from eligibility to run for the board.
The extent of the power of the HOA is limited to the controlling laws of the State/County/City where the property is and whatever legal restrictions the HOA has in writing. Many of these HOA's are little power clubs who enjoy ruling over their neighbors. I would recommend you arm yourself with the HOA rules and regulations and if there is no prohibition for residents to serve based on the home being held in Trust (it would be a first for me, and I represent a couple of HOA's), than unless otherwise prohibited, I would think you wife can run.
This answer is marginal legal advice and does not constitute an attorney-client relationship. Every client and case is unique. The best advice is to always consult with an attorney. Free legal resources at www.ZippToCourt.com