The well-known repeater listings/apps don't provide a database that you can download and use freely for any use in building your own app.
Is there a such a freely accessible repeater listing database online for amateur radio?
Since there was no such repeater listing out there, I started my own database:
As a bonus, there is a service to comment on, and favorite to get status notifications about a repeater, something I haven't seen in other listings. Also, a topo map showing where each of the nearby repeaters is, is now available in Repeater-START for Linux.
There are several web sites, mobile phone apps, and printed books offering repeater lists. The hearts of these sites, apps, and books are surely databases. However, as @K7PEH says, collecting the information and making it easily available to the public costs time and money. Every example of such a site, app, or book that I've seen is trying to make money from the information, if only from advertising. I could be wrong, but I doubt that anybody would give away such a database for free, or keep it updated for free, at the present time.
There is a JSON formatted data base at: https://www.radioid.net/static/rptrs.json This database is accessible from the main page by clicking on 'Database', then 'Database Dump Files', so I assume it is intended for public viewing. I don't know contents of these files or the license, but the Chirp software uses an old broken link for this repeater database.
I've been using RepeaterBook. There's an app for Android or iOS, as well as a web site accessible with any modern browser. The app is free, to download and to use, though both the app and the web site request donations. Repeaterbook lists 10m, 6m, 2m, 1.25m, and 70cm repeaters in FM, DMR, and possibly other modes, and is searchable by band, call sign, or distance from a location. It also includes notations for networked repeaters and those that support services like Echolink.
RepeaterBook is also an option for Chirp software to automatically obtain repeater lists, and I've used it that way as well (when programming my mobile for the repeaters along my usual route to a coastal vacation spot).