Re: "can I have people signup, deposit money with me, and purchase stocks through my account ... ?"
Short answer: No, that wouldn't be legal.
But anyway, perhaps refer to SEC.gov - Guide to Broker-Dealer Registration. Section II A defines a broker as "any person engaged in the business of effecting transactions in securities for the account of others". In your idea, as it is stated, you would be a broker and would need to register as a broker. I don't think passing through to a broker would get around that.
But let's say you do register as a broker. The next problem with what you've laid out is that you can't purchase stocks for other people through your account. See, for instance, SEC.gov - Investor Bulletin: Custody of Your Investment Assets. Here's a relevant part, with my emphasis:
First, with certain limited exceptions, an investment adviser is
required to maintain client funds and securities with a “qualified
custodian.” Qualified custodians can be banks, registered
broker-dealers, futures commission merchants, or certain foreign
entities. A qualified custodian either maintains client funds and
securities in a separate account for each client under that client’s
name, or in accounts that contain only client funds and securities
under the name of the investment adviser as agent or trustee for the
i.e. Wherever you would hold assets on behalf of clients who use your trading/investing app, it couldn't be your account — it would need to be their account, with controls in place to ensure that you couldn't abscond with client funds to a non-extradition destination (tropical or otherwise).
Ultimately you can _kind of_ do what you're asking. Robo-advisors are examples that come to mind: many have apps for clients, and robo-advisors execute trades on behalf of clients. As with those services, you'd need to follow established regulations, and the two examples mentioned above are just the tip of the iceberg. I'd wager that building your app would be the trivial part of any such undertaking. Maybe find a partner who is already a broker with access to custodian accounts? And a good lawyer.