Calendimaggio Assisi?

It starts the first Wednesday in May.

Cannot answer the "tickets" issue, but this seems to answer the 'when'

The “Calendimaggio” Festival
Lasting three days, for decades the “Calendimaggio” Festival was held from April 29th to May 1st, to maintain the ties with the medieval May Day Celebrations. However, for several years now, it has been decided to celebrate the Festival on varying dates, to avoid the “Calendimaggio” coinciding with other secular celebrations, and it *now takes place on the first Wednesday, Thursday, Friday and Saturday of May, May 1st excluded.***

Above found on this site-- https://www.calendimaggiodiassisi.com/en/programme/calendimaggio-open

Thanks for the quick replies. We will be there the first Wednesday!

In 2020, the main activities should be Wednesday, May 6 thru Sat., May 9. (There are other "lesser" events that happen in the days before the official opening.) I was fortunate enough to be there for the entire festival in 2015. I have memories (and tons of photos) that I will cherish until I die. It is a photographer's dream. Check out the photo archives to get an idea.
https://www.calendimaggiodiassisi.com/en/archive

You'll need to buy tickets for almost anything that takes place in the main piazza, Piazza Del Comune, or be relegated to being crammed in with the crowds on foot at the opposite ends of the piazza. The seating is very limited, in a set of metal bleachers that get set up opposite the Chiesa di Santa Maria Sopra Minerva (the one with the ancient columns). Tickets will be available on that same website, but not until much closer to the actual festival. Just keep checking. It does sell out.

A few things to keep in mind:
1. This is a festival done by and for the locals, meaning little if any accommodation is made for tourists during this time. Expect streets to be closed down and access to be limited in lots of places around the historic city. The only language you will hear over the speakers during the festival is Italian. No English. Ever. Prepare to be utterly flumoxed at what's going on.
2. Some of the events go VERY LATE into the night. In particular, on the preceding Monday and Tuesday nights, there are two separate "Scenes in the open air" that start at 9:30 and are basically a moving play that travels throughout a large part of the city. This used to be available free to the public but I think they've restricted it to just the judges now. Then on Wednesday and Thursday there are recreations of medieval life. These are free, but you HAVE to be in costume to get in, and there are no costume rental places to be found. (This isn't Venice.) The earliest those start is 9:30 pm, but more likely not until 10:30 pm. They easily go until 1 a.m. and can be very rowdy at the end. Also, the awards ceremony that final Saturday night will be party central for the winning side.
3. You might want to avoid staying in the immediate area of the three main piazzas: Piazza Del Comune, Piazza San Rufino (for the upper town) and the area in front of the big green lawn in front of the Basilica (where the lower town celebrates).
4. Italy is one big safety code violation, so don't be surprised to see fire and open flames in the evening in places that would scandalize a US Fire Marshall.
5. Stay in town, not outside of town. Ask your hosts/hotel manager for tips on how to enjoy the festival. Arrange for your lodging NOW.

Have a great time!