I will second @StephenS's image as the best you're likely to find with minimal effort. In fact there are a few of these images available if you do an image search for "FAA STARS TAMR" and I've included all of them at the bottom of this answer.
If you need the specific boundaries you can submit a FOIA request. If you're a private citizen who wants the information for non-commercial use this should be free, but it may not be. Here is the FOIA form for the Air Traffic Organization FOIA Coordinator for the Eastern Service Area, which should be who you want. I would probably say you're requesting "A document showing the boundaries of Fayetteville ATCT center-delegated airspace" and that you think it would be found in the SOP and/or in the LOA with Washington ARTCC.
Alternatively, if you have a controller friend who's willing to do you a favor you could ask them to pull up the SOP/LOA for FAY from the FAA intranet next time they're at work. All facilities should have their SOPs and LOAs stored on a system that can be accessed at loa.faa.gov from behind the general firewall. This is probably against policy though, so they might not be willing to do it for you.
Unofficial but likely workable answer
The VATSIM community strives for realism, and from time-to-time submits FOIA requests to the FAA to form the basis for their SOPs and maps. The Virtual Washington ARTCC website includes a page of downloadable "facility files" that you can open in a virtual radar client in order to see the airspace boundary (by enabling the "VIDMAP" map). I have no reason to doubt this map's accuracy, as it looks quite detailed and appears to match the boundaries shown in the maps below, but there is no guarantee it is accurate or current.
To find the geographical coordinate of any given point on the map, you can type
<F7>D* and click on the point. This shows the coordinate in what appears to be a hybrid Degrees Minutes.TenthsHundredths format. These coordinates must be defined somewhere in the downloaded "facility file" but I do not see a way to extract them on their own; you could get in touch with the VATSIM community to ask for the originals.
This method would probably work for any facility in the country, and although there is no guarantee of accuracy, airspace boundaries do not change often at all. It may be more time-effective to ask the various VATSIM communities for access to their FOIAed documents than to duplicate their effort by asking the FAA for each individual facility.
Interesting graphical history
An older map, still shows ARTS facilities. No date. FMH and ACK still separate. NMM is an FAA facility. JST not shown. Original
Map from September 2013. FMH and ACK still separate. NMM is an FAA facility. JST not shown. Original, 1620x1080. This is the highest-quality map of the bunch.
Map from September 2014. Very poor quality, hard to tell the differences from 2013. FMH and ACK still separate. NMM is an FAA facility. JST not shown. Original
StephenS's undated map. FMH and ACK have merged to create K90. NMM is an FAA facility. JST not shown.
Map from July 2018. This version shows ARTCC boundaries as well! K90 has been absorbed by A90. NMM is now run by the DOD. JST is a wholly new approach facility running airspace delegated by ZOB, located between PIT and MDT. The facility covering Edwards AFB and the R-2508 complex is now designated E10, "High Desert TRACON," instead of EDW. Also shown are the then-in-progress airspace consolidations (Section 804 or otherwise): MFR to EUG, YKM to PSC, MSO to GEG, RNO to SCT, BZM to BOI, GJT and PUB to D01, SJT to MAF, FYV to FSM, COU to SGF, LYH to ROA. Original
Map from June 2019. STARS is almost ubiquitous, with only two ARTS 2E and four ARTS 1E facilities left that I can see. Transfers of YKM to PSC, COU to SGF, and LYH to ROA are complete (as are the others from 2018, I think, but the arrows are still drawn because the airspace is not contiguous). MKG, MBS, LAN, and FNT have been combined into a super-TRACON at AZO now known as "Great Lakes Approach." Original
As of 2021, I believe there are no remaining ARTS facilities; all facilities are using some flavor of STARS. E10 is now JCF, "Joshua CCF." MFD and CAK have both been consolidated to CLE. PSC is being consolidated to GEG, taking YKM airspace with it. PIA and SPI are being consolidated to T75. HUF is being consolidated to IND. There are probably some more I'm not remembering.