According to Scooby-Doo Behind the Scenes (released in 1998; see Wikiquote), the initial money to start out was given to them by Daphne’s dad:
Daphne: "Well, Daddy gave us the money to start off. (shows an image of the check that first financed Mystery Inc.) We didn't even have a car, so we gave our parents gas money to drive us around."
In addition, many of the different canons have shown them having jobs. (In some of these they are adults though.) From the Wikia page for Scooby-Doo:
- Junior detectives-for-hire: Contrary to the above, when the gang was in their preadolescence, they worked out of a clubhouse, calling themselves the Scooby-Doo Detective Agency, and charged a minimal fee for their services in solving the mysteries in their hometown of Coolsville. (APNSD)
- Actor: Scooby got the role of Sandy Duncan's dog, after solving the case at Mammoth Studios. (TNSDMovies: Sandy Duncan's Jekyll and Hyde)
- Carnival owner: Scooby, along with the rest of the gang, was given a share of Dick Van Dyke's carnival, after saving it from going under. (TNSDMovies: The Haunted Carnival)
- Construction worker: He, Shaggy, and Fred took questionable jobs at a construction site. (TSDS: High Rise Hair Raiser)
- Freelance journalist: Scooby, Shaggy, Scrappy, and Daphne became freelance journalists, forming the second incarnation of the Scooby-Doo Detective Agency. (TNS&SDS, TNSDMysteries)
- Gym teacher: He, Shaggy, and Scrappy were hired as gym teachers at Grimwood's Finishing School for Girls, or so they believed. (Scooby-Doo and the Ghoul School)
- Co-racecar driver. (Scooby-Doo and the Reluctant Werewolf)
- Airport customs: He and worked in customs at an airport during the break-up of Mystery Inc., but were fired when they ate all the confiscated cheese. (Scooby-Doo On Zombie Island)
- Detective: He, Shaggy, and Scrappy worked for Shaggy's uncle Fearless Shagaford at the Fearless Detective Agency. (The Scooby & Scrappy Doo Puppy Hour)
- Additionally, Scooby assisted Shaggy when he was a paper boy, by unloading the delivered papers into his bike's basket. When the gang discovered Shaggy's boss stole his bike chain to work his counterfeiting machine, Shaggy quit, and so did Scooby. (APNSD: A Bicycle Built for Boo!)