I wouldn’t worry if it’s only been a couple of hours. Don’t forget they may have little opportunity to recharge phones, or only certain times when they have any phone signal. They are of course 2 hours ahead of us.
If they and you are happy to proceed then yes that is when the visa application needs to be done. I assume you have already registered on the governments Homes for Ukraine scheme. The visa applications are made through that. This can be done by them (in which case you will need to provide them with information about you, including a pdf copy of your passport), or by you. Most people I know who are hosting fill out the forms on behalf of their guests. It’s a very long winded form which is difficult for us even as English speakers. I’m not sure how anyone trying to do this in an unfamiliar language, on a mobile phone would cope. One application needs to be made per person.
We completed the forms at a time prearranged with our family in Ukraine as there were lots of questions we didn’t know the answers to. We WhatsApp messaged them throughout the process. They will also need to provide you with photos of their passports and if possible something like a utility bill with a date and their address on it to prove that they were living in Ukraine on 31 Dec 2021 (although we didn’t have this for our family). You will need to convert these to pdf format.
I hope this info helps. It’s a long road ahead of you, we are still waiting for visa approval having applied on 23 March.
I hope this is if some help. Good luck.