Does RDRFR work in RDR Online?

No, RDR First Response is not compatible in any way with RDR Online.

RDRFR is made possible by the RAGE Plugin Hook, a modding tool which doesn't work in RDR Online.  This is because of two main reasons:

1. We do not wish for people to use the mod's features in public RDR Online sessions as this could lead to griefing.
2. Rockstar Games prohibits any modifications which interfere with RDR Online.

Quote

Rockstar Games believes in reasonable fan creativity, and, in particular, wants creators to showcase their passion for our games. After discussions with Take-Two, Take-Two has agreed that it generally will not take legal action against third-party projects involving Rockstar’s PC games that are single-player, non-commercial, and respect the intellectual property (IP) rights of third parties. This does not apply to (i) multiplayer or online services; (ii) tools, files, libraries, or functions that could be used to impact multiplayer or online services, or (iii) use or importation of other IP (including other Rockstar IP) in the project; or (iv) making new games, stories, missions, or maps. This is not a license, and it does not constitute endorsement, approval, or authorization of any third-party project. Take-Two reserves the right to object to any third-party project, or to revise, revoke and/or withdraw this statement at any time in their own discretion. This statement does not constitute a waiver of any rights that Take-Two may have with respect to third-party projects.

You can find out more about Rockstar Games' modding policy here:

https://support.rockstargames.com/articles/115009494848/PC-Single-Player-Mods