So I just went on the website now after trying multiple times when the site initially 'opened' and after taking a break to do some gardening and I was able to order the Minnie plush. As soon as my ordered was processed, I went back to check and it is now sold out. I am doubtful that I will actually receive the item and will mostly likely get an email from
Shop Disney indicating that the item is unavailable for shipping or something like that

.
ETA: Okay, so I checked again and it is saying that the mug and plush are available. Either they are releasing the items in waves or this is likely just another website issue. I think I will stick with my original hypothesis that I am probably not going to end up with the plush after all...lol.