A petition, by itself, likely would not be legally binding. Florida has
separate statutes for HOA's and condominiums. Also has process for
arbitration, but that is generally for administrative wrongs (board has
not complied with statutes or bylaws), not to take issue with a
particular vote.

I'm in Michigan. The only thing the petition was for was to call a
special meeting, and we followed the guidelines which says this is
allowed in the bylaws.

I agree the board had the legal power to pass the assessment off in the
budget, but a meeting can be called to 1) amend the bylaws (so that
can't raise the budget to whatever they want, for example) or to 2)
recall board members. We haven't even gotten that far.

Having enough signatures to call a special meeting doesn't meet muster
here, as the board still has the authority to vote for maintenance and
repairs, not the association.

Why would that matter? The meeting could be for anything.

You may have enough signatures (members
to vote) to recall board members, and a new board could then vote down
the project (unless contracts already signed).

That's the plan. The contract is not yet signed.

In Florida, the members
have to vote if there is a material alteration to the common areas -
like installing a new pool - but repairing existing roads would not seem
to be a material alteration. Read your documents. Members can always
sue here, but in FL the loser pays costs.

I'm in Michigan, and have had trouble finding laws regarding my state.