Ambition is a hybrid (F1) so if you save seed, you will not get Ambition but a genetic roulette of traits. No harm in trying if you are curious, don't rely on it for your yearly crop though.
Shallots are like onions, if they are started from sets, they will act like a biennial and send up a flower spike if they were big enough sets. I just break off the flower spike and the bulbs develop normally. BTW, I grow Ambition also, from both seed and from the previous years' sets for the REALLY big shallots I use to impress my friends LOL.

I started Ambition Shallots from seed last year and planted a couple of them in the fall when I planted garlic. They came up looking strong but they got the flower bud in early June this year. I pulled the flower bud but they didn't bulb up at all.

I planted about 15 Ambition Shallots that I had from last year in April this year and they are amazing! Almost all of them have divided into 5-6 very large shallots. I plan to keep some of them to plant next spring.

As a side note, the shallot sets I purchased for planting this spring did not divide that well and are not as large as Ambition.

Try planting your sets grown from seed in the spring.

A little update on the shallots that I planted in the fall. It seems that they dug themselves deeper than I had planted them and they do indeed have some very nice size bulbs.

So, to increase your stock of Ambition, just keep planting the sets from each year. Don't bother trying to grow the seed because you just don't know what you will get. the sets are a sure bet because they are a clone of the plant.

Good Luck!