Indiana House Democrats say in a letter the proposal could cause "dire" results in Indiana.
The Indianapolis Star reports that Pence communicated with advisers through his personal AOL account.
The state supreme court hasn't yet decided whether it will consider the appeal.
Former Gov. Mike Pence denied the request for emergency status before he left office.
Pence began his political career in Indiana nearly two decades ago.
The appointments include one man who kept a lucrative state lobbying contract while he also had a paid position with the Republican presidential campaign.
Tickets for the Republican National Committee fund-raiser started at $2,700 per person.
Keith Cooper has asked Gov. Pence for a pardon based on evidence he was wrongly convicted. Pence says he must first exhaust all legal options for exoneration.
Pence said Monday that he believes Trump is sorry for the crude and predatory language about women captured in a 2005 video that became public last week.
In a statement published on his Twitter account, Gov. Mike Pence condemned Trump's words, saying he was offended by them.