So Assistant Head Coach/Co-Defensive Coordinator/Defensive Line Frank Maile is the interim coach. USU announced they will run a national search for a replacement.
This is who
KSL.com claims could be candidates:
Jay Hill - Head Coach, Weber State
Ties to Utah State: Played cornerback at Utah from 1998-99, coached college football in the state of Utah since 2001
Hugh Freeze - Head Coach, Liberty University
Ties to Utah State: None
Rich Rodriguez - Former Head Coach, Arizona
Ties to Utah State: None
Jeff Grimes - Offensive Coordinator, BYU
Ties to Utah State: Coached college football in the state of Utah for six seasons
Ed Lamb - Assistant Head Coach/Special Teams Coordinator, BYU
Ties to Utah State: Played defensive end at BYU from 1994-96, coached college football in the state of Utah for 14 seasons
Robert Anae, Offensive Coordinator, Virginia
Ties to Utah State: play offensive line at BYU from 1981-84, coached college football in the state of Utah for 11 seasons
Morgan Scalley - Defensive Coordinator/Safeties Coach, Utah
Ties to Utah State: Played defensive back at Utah from 2001-04, coached college football in the state of Utah since 2007
Frank Maile - Interim Head Coach/Co-Defensive Coordinator, Utah State
Ties to Utah State: Played defensive line at Utah State from 2004-07, coached at Utah State for 10 seasons
Rocky Long, Defensive Coordinator, New Mexico
Ties to Utah State: Born in Provo, Utah, served as head coach in Mountain West Conference for 20 seasons
Kelly Poppinga, Linebackers Coach, Virginia
Ties to Utah State: Played linebacker at Utah State from 2003-04 and at BYU from 2005-07, coached at BYU from 2009-15
Jim Harding, Assistant Head Coach/Offensive Line Coach, Utah
Ties to Utah State: Coached at Utah since 2014
David Yost, Offensive Coordinator, Texas Tech
Ties to Utah State: Served as Utah State’s Offensive Coordinator from 2017-18
The Athletic also reportedly mentions Aaron Roderick as a candidate but it is behind a paywall so I won't link it.