As PP mentioned, you can but you're technically supposed to pick it up at Old Port Royale, the main entrance. However, I know that for Uber/Lyft rides, they will pick up and drop off at the various villages. So check the Uber Eats app to see if your village location is there. If so, you should be able to meet them at the drive-up circle for your village where the sign is. We didn't use Uber Eats, but we did order Lyfts to Jamaica several times and it was no issue.
Glad everything worked out for you. For the benefit of others reading this, its best to call to have room requests noted at least a couple days in advance as room assignments are typically done a couple of days out from your check in date. If they originally assigned you Martinique, it sounds like they had upgraded you to a king preferred? What building did you end up in?