According to Detroit News, there's precedent for the state board to certify results that a county board did not:
The Wayne County Board of Canvassers failed to certify the 2013 mayoral primary candidates within 14 days and left the decision to the Board of State Canvassers. The Michigan Bureau of Elections reviewed the ballots, and the state board eventually certified the results and found write-in candidate Duggan won the primary.
This is of course legal, as the first part of Rick's answer explains.
Note that the Michigan state board has not yet certified the results in the 2020 presidential election in the state, and the RNC has requested them to "pause" the certification.
In a letter sent Saturday, Republican National Committee Chair Ronna McDaniel and Michigan Republican Chair Laura Cox urged the [state] board to adjourn for 14 days to allow a full audit and investigation so "numerical anomalies and credible reports of procedural irregularities" can be addressed.
The state board is also 2:2 split between Republicans and Democrats.
Presumably, the ultimate goal of Trump in Michigan (although the intermediate steps might differ) is similar to what was revealed in a Pennsylvania lawsuit by Trump's team, which
asked the judge to declare Pennsylvania's election results "defective" and let the Pennsylvania General Assembly choose who gets the state's 20 Electoral College votes. The General Assembly, unlike the popular vote in Pennsylvania, is majority Republican.
However, insofar leading Republican legislators of Michigan (who recently met with Trump) don't seem enthusiastic about that prospect:
"We have not yet been made aware of any information that would change the outcome of the election in Michigan and, as legislative leaders, we will follow the law and follow the normal process regarding Michigan’s electors, just as we have said throughout this election," Chatfield and Shirkey said.
There's now a long CNN article on all the "what ifs". The gist seems to be that the present Michigan laws don't give the state legislature a direct override. And the Democratic governor of Michigan could block a last minute legislative attempt at changing those laws.
As a more recent development, the Michigan state board certified the election; one of the two Republicans on the board voted for certification, while the other abstained. Coincidence or not, roughly on the same day Trump's GSA appointee announced they'd begin the transition to a Biden presidency.