I've no idea how the Raymarine works, but if it accepts NMEA0183 "commands"/signal however you want to call it, then yes, all Garmin plotters would be able to send instructions to the autopilot.
Should work fine, I run my Lowrance AP from my Raymarine plotter. You just have to study the wiring diagram to connect the correct two nmea wires. OUT to IN and IN to OUT and make sure you choose the correct baud rate as Raymarine can talk to other units on two different date transmission speeds, its very simple to setup in reality.
I run an older Ray A/P from my Garmin plotter. There were some teething issues which IIRC were to do with waypoint names being too long or some such but all seems fine now.
Be very careful - while it is certainly the case that it should work, my experience is that it may be unreliable. I installed the previous generation of Raymarine autopilot with Lowrance plotters on our previous boat about four years ago and they never reliably played nicely together. I tried them linked by both NMEA0183 and NMEA2000 with equally unreliable results. More than half the time, I would power everything up and the plotter would simply fail to recognise that there was an autopilot present on the network. On those occasions when it did see the AP, it would work for a while, but then try to steer me off in random directions which could easily include onto the beach.
I spoke to both Raymarine and Lowrance (Simrad) and neither was particularly surprised that they were not compatible with the other. Each blamed the other for not publishing enough information about their products to permit competitors to ensure compatibility. When we sold the boat, they had never reliably worked together - we installed a 100% Raymarine solution on the new boat and it has worked perfectly.
Ah, that does put a rather different complexion on it - you'll always have the option of the Raymarine if you hit compatibility problems with the Garmin.
It occurs to me that some of the earlier posts were referring to NMEA0183 - do bear in mind that the current generation of Raymarine APs are NMEA2000 (or, better, Seatalk HS) as their preferred protocol - I'm not sure if they even include NMEA0183 natively, or if there is a need to install a converter if you want it. Both your E80 and the Garmin will support NMEA2000, and I would try that first - it's much easier to wire, and the functionality will be better provided the Garmin and Raymarine agree to talk to each other.
You are intending to fit a Raymarine AP control head at the binnacle, aren't you? Raymarine used to insist on it, even if you also had a full specification plotter there. These days, they do seem to accept the option of relying on a modern Raymarine MFD as your only AP control, but I'm not sure I would rely on a non-Raymarine device - the consequences of being unable to disengage your AP at short notice from the helm could be unpleasant!