I've also taken many boxes at once, so there should be no problem for them. They do have reasonable guidelines on what they'll accept.
They are delighted to get your donations, however I have seen times when someone came with a truck full and they did not have space for it all. I helped a friend who had a lot to get rid of we took it in three trips and had no problem.
I have seen a couple of storage units outside and I think they have a pretty good sized receiving area too. I don't work there, but I've been there to shop many times. I'm sure they can take wherever you have and will be very happy to do so!
As much as you can. It's really appreciated
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