Signet is a Bitcoin testnet whose blocks are not created per proof of work, but rather, they are signed into existence by the parties running the Signet. Therefore, it is not possible to create your own blocks on Signet.
A faucet itself is not a terribly difficult software project. It would need some way of users inputting their address, an attached wallet, and an automated process that sends funds to the requesters.
To get the faucet funded, the operators could probably convince the Signet operators to provide signet coins. The #bitcoin-signet channel on the Libera Chat would probably be a good contact point to coordinate with the Signet operators.
The biggest challenge with operating a Testnet/Signet faucet these days appears to be that some people have started trying to monetize access to Testnet/Signet coins, a classic tragedy of the commons.
Instead of being a free and accessible developer tool, Testnet/Signet coins have become scarce and are traded altcoin trading platforms as these people monopolize block production on Testnet and hammer faucets to suck up all coins. This has in turn increased the workload on faucet operators, who now have to rate limit coin distribution, vet requests, and shore up DoS resistance of their faucets.
Some faucets have turned to authenticating faucet users, e.g., per their Twitter account, requiring such accounts to have some minimum of activity and limiting the payout per time period.
Alternatively, projects that significantly rely on Signet for experimentation could simply run their own custom signet and distribute coins on their own signet to their own users.
