Jennifer Burnes says the paychecks from her work as a model and designer in Hollywood arrive sporadically, posing a challenge to paying monthly rent. “You know the check’s on the way, you just don’t know when, exactly,” the 22-year-old said.

That is why she turned to StayTony. The property manager has teamed up with Uplift, one of several startups offering loans to recent college graduates, professionals moving to a new city and others who want to build credit or could use assistance making rent payments.

These...