Yes, your puppy will likely need to repeat some of the initial vaccinations if they become very overdue. It will vary somewhat by your location as to what vaccinations your veterinarian recommends.
It is important that puppies receive their vaccines on time as their immune system is developing, and vaccination when young affords them the best protection as they get older.
AAHA has produced the 2017 AAHA Canine Vaccination Guidelines on dogs Overdue for vaccination. Here you can work out exactly which vaccines your puppy may need to restart based on what he has received already.
I would encourage you to see if any other clinics are open locally to vaccinate your puppy. Generally puppy and kitten vaccines are considered time sensitive, so many clinics will still be offering them. For an older animal it should usually be fine delaying vaccines a few months. My clinic, and most others in the area, are now collecting the animal from the car, then performing the exam, vaccines, treatments, etc. and discussing with the owner over the phone, to minimise contact between people.