I started to work today. I started to work today.

This is correct, but it would sound more natural to say, "I returned to work today." The meaning is also clearer.

I startedwent back to work today. I went back to work today.

I'm replying to inquiries received during new year’s vacationthe winter break. I'm replying to inquiries received during the winter break.

"Winter break" might just be an American way of putting it. You could also say "during the holidays" or "during the vacation." New Year is capitalized. Present progressive ("I'm replying") makes it clearer that you're doing this right now.

I replyied to inquiries received during the new year’s vacationholiday. I replied to inquiries received during the new year’s holiday.

Now iIn Japan right now, the infected Covid-19 cases are increasing, so no customers coame to our shop. In Japan right now, the infected Covid-19 cases are increasing, so no customers came to our shop.

However, I am very grateful to have received those inquiries. However, I am very grateful to have received those inquiries.

I hope everybody is satisfied with my replyies. I hope everybody is satisfied with my replies.

This is a super small thing, but "reply" suggests that you sent the same message to everyone, while "replies" suggests that you responded to each inquiry individually.