Is there a software that can detect if brand ambassador's social media posts result in app downloads?
You could use unique UTM Codes in the links you require each ambassador to use and track the traffic that way.
That's a complex question and it has more than one moving part. Yes, you can track almost any digital transaction, but the best methods to do so aren't always straightforward.
Share them a link with a unique referral code or ask them to share a specific code with their users in exchange of some reward/contest etc. Can you elaborate a bit on what type of social media post are you asking them to post?
I don't think you need a software product. Providing a unique link to each should do it like EF says. They are currently providing a link now I assume.