For house seats, federal law requires an election.
For Senatorial seats, states can decide whether to have special elections or have the Governor appoint someone to fill the seat until the next general election.
The examples you reference from the Trump administration are all House seats, I believe.
Arizona allows its governor to make the appointment, with the proviso that the appointee must be from the same party as the person who vacated the senate seat.
For more on this see: http://www.ncsl.org/research/elections-and-campaigns/vacancies-in-the-united-states-senate.aspx