How much time do I have to wait for coins? I sent from the website to the stock market wallet. It's been an hour and no coins!

Not sure haven't used that service in a while... I've been buying my ion from Bitrex
I'm assuming you mean ionomy.com. If you contact them with your details they will be able to help

I bought ION on bittrex and sent to ionomy.com the transaction lasted only 5 minutes. When I sent ION from ionomy.com to bittrex - no coins and no transactions

It's 6 confirmations so the same transaction time of a few minutes max
Your problem has effected a few other people too. Just seen on discord that the hot wallet has run out of ION and is now in the process of refill so expect normal service to be resumed shortly.
If you haven't already join discord as their are people there that can help..... sometimes it's quicker to ask in discord than sending a mail to support
https://discordapp.com/invite/vuZn7gC