None of the banks operating in Estonia currently open business accounts to e-residents without an in-person visit. You can find the list of banks here.
If you can visit Estonia we would suggest applying for an account with LHV Bank online here. You can apply only after you have registered your company.
If your online application is approved you can book your travel to Estonia to finalize the account opening process.
Alternatively you can open business IBAN with one of the licenced fintech companies such as TransferWise, Payoneer, Neat, Paysera, Revolut or apply with your local bank at your location. Read more here.
It is important to check what type of proof of address may be required to prove your company location. You can find our advice here. TransferWise does not require proof of address from Estonian limited companies.
Please be sure to choose a service provider that can issue a digitally signed proof of share capital deposit. TransferWise is capable of issuing share capital deposit certificates with the necessary digital signature. You would need this document when registering your share capital.
If you are looking for online payment acquisition via merchant account please see this article.