Open Ephemeris
Sub-arcsecond planetary positions, natal charts, transits, and Human Design from NASA JPL DE440

Catalog of 16,000+ satellites merging CelesTrak, GCAT, Wikidata; free tier included.
OrbitalWiki API is a free Science & Math API that requires apiKey authentication, is served over HTTPS and supports CORS, so it can be called from the browser.
Free REST API for satellite orbital data, TLE/OMM elements, operators, NORAD and COSPAR records, with field-level provenance where available.
Details from orbitalwiki.com.

Unified API suite · now live
Geocode an address, validate an email, track a flight or pull stock data with one account, one dashboard and one API key.
Sub-arcsecond planetary positions, natal charts, transits, and Human Design from NASA JPL DE440
3D structures of proteins and nucleic acids
Repository and archive for study designs, research materials, data, manuscripts, etc
The OrbitalWiki API is provided by OrbitalWiki (orbitalwiki.com). Free REST API for satellite orbital data, TLE/OMM elements, operators, NORAD and COSPAR records, with field-level provenance where available.
Yes. The OrbitalWiki API is listed in the Science & Math category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
Yes. The OrbitalWiki API uses apiKey authentication, so you need to register for credentials before making requests.
Yes. The OrbitalWiki API supports CORS, so front-end JavaScript can call it directly.
Yes. The OrbitalWiki API is served over HTTPS, so it's safe to use in production.