I think this is sort of ok to some extent, if you have ordered everything you need. You have a screenshot of the order and your account has been charged, so I’d say the order was likely registered properly.
However, it is alarming that you can’t log in and don’t get any e-mails from the shop. This might cause problems with delivery notifications down the line. So think this is one of the rare instances where you should send Jolla an e-mail.
Use a Subject line that indicates the problem and include as much detail as you can. Since you don’t have a confirmation e-mail to reply to, you should probably include the screenshot and date/time when the order was completed (the file timestamp of the screenshot should be close, if you have nothing better).
Also, please have a look at the FAQ for more information that may apply.