I think things are in flux with the Tsum Tsums at this point. A few months ago, it was not uncommon for some of the newly released designs to sell out within the first 10 minutes (or less) of being released. Not all designs, just select ones. For example, all but three of the Peter Pan designs sold out in the first 20 minutes. The Marvel designs have all lasted much longer, so either they produced more of those or they were not as popular.
For the last few releases (e.g., the Lion King, the Valentine Box Set, Zootopia, and the birthday pair) most of the designs have been lasting a lot longer. For example, you can still get 2/3rds of the Lion King characters (Scar, Ed, and Musafa sold out early), all but one of the Zootopia (the Sloth), the Birthday ones, and the Valentine's Day set. My inclination is to say that they've really increased production of the Tsum Tsums so they'll probably be more readily available in the future. I would also guess that there will continue to be one or two designs per set that sell out quickly (although, I don't think quickly is going to be as quick anymore).
In terms of the website, the site can be temperamental right when the new Tsum Tsums are released because of high traffic on the servers. I recommend logging in before they're released if you're going that route. The new Tsum Tsums are released somewhere between midnight and about 12:30 am Pacific Time (it's not consistent, but it's somewhere within that window generally).
I've had items removed from my cart (a note saying they were now sold out or unavailable), but I've also been able to go through with the order and then got an email about a day later saying of the items were unavailable. So unfortunately, the only way to know if you really got the item is to wait until you receive the emailing saying your items have shipped.