My provider supports this in more expensive plans, however it seems that I found solution, which will allow to automate whole process. My provider disabled automatic Let's Encrypt certificate issuance, but I can still use DirectAdmin API and cron, so I can just do couple requests to DirectAdmin to automatically add acme-challenge TXT records, remove them after verification and also automatically add new certificate. Now I just need to find the best PHP client for this job (there's 4 with ACMEv2 support) and I will be ready to get SSL certificate for my domain and fully automate it. 
Thanks for your help! Now I know, that I should renew certificates at about 60 days, in case somethiing bad happens.