Below is the list of NCAA appearances since 2010 (so 14 is the highest number possible, as I counted 2020 bids, as appearances) of schools that also field a FBS program. To be clear, this is CURRENT FBS programs, so some of these appearances were before they actually had a FBS program, but that was WAY too much work to sift through for me.
When you are combine football success, the ones that really stand out are: SDSU, Cincy, UH, Memphis, BYU and Boise. Coincidentally, those are programs that are either headed to P5 status or are always in the discussion.
San Diego State MWC 10
Cincy AAC 9
NMSU Indy/WAC 9
U Conn Indy/Big East 7
Houston AAC 6
Memphis AAC 6
Temple AAC 6
Utah State MWC 6
BYU Indy/WCC 6
Buffalo MAC 4
Boise State MWC 4
New Mexico MWC 4
UNLV MWC 4
Nevada MWC 4
Ga State SBC 4
MTSU CUSA 3
UAB CUSA 3
Ohio MAC 3
Akron MAC 3
Colorado State MWC 3
ODU SBC 3
Liberty Indy/ASUN 3
SMU AAC 2
Tulsa AAC 2
NTSU CUSA 2
WKU CUSA 2
Kent St MAC 2
Wyoming MWC 2
Coastal Carolina SBC 2
Louisiana SBC 2
USF AAC 1
FAU CUSA 1
UTEP CUSA 1
W Michigan MAC 1
Fresno State MWC 1
Hawaii MWC 1
James Madison SBC 1
Marshall SBC 1
App State SBC 1
Troy SBC 1
USM SBC 1
U Mass Indy/A10 1
Tulane AAC 0
UCF AAC 0
Navy AAC 0
E Carolina AAC 0
UTSA CUSA 0
Rice CUSA 0
FIU CUSA 0
Charlotte CUSA 0
La Tech CUSA 0
BGSU MAC 0
Miami Ohio MAC 0
Toledo MAC 0
E Michigan MAC 0
Ball State MAC 0
C Michigan MAC 0
N Illinois MAC 0
Air Force MWC 0
San Jose State MWC 0
Ga Southern SBC 0
So Alabama SBC 0
ULM SBC 0
Texas State SBC 0
Arkansas State SBC 0
Army Indy/Patriot 0