Forget the 120v dorm fridge in a Minivan without hookups. Just forget the idea ever occurred.
Also don't plan on being a chronic battery abuser/killer/returner. Trying to beat the system screws everybody.
Bite the bullet and get a Waeco chest type fridge with a Danfoss compressor designed to run on 12 volts. 100 watts of solar will have little issue keeping up with the draw of this fridge.
Do take advantage of alternator recharging too, but don't, as so many do, think they are some instant magical battery recharger. An Additional battery tacked on to the charging circuit over long thin wiring will not receive much from the alternator. An additional battery heavily depleted, hooked to the alternator over Fat short cabling, can fry the alternator as they were not designed for such a task.
If you are planning on putting this battery in the interior of your miniVan, then you want an AGM battery. A regular flooded battery needs to be ventilated to the exterior of the vehicle as it will offgas during charging, and you don't want those fumes inside.
Do make sure the battery cannot move, and the cables are protected from Chafing, and if you do get a Waeco Chest fridge with a Ciggy plug cable, run a dedicated ciggy plug right to and fused at the battery with 10awg cable as per the Danfoss controller recommendations.
Do not use a starting battery.
Trying to save money initially on this by a dorm fridge and a starter battery will cost you more in the long run, with a whole bunch of hassle and drama in between.