I would like to put together a web app allowing users to get the full list of apps/software they installed during time using their Apple ID. I won't be storing their passwords or anything.
Is there an Apple API method for this?
This is not possible. The Apple API is a search API for the iTunes catalog. You can only retrieve informations about applications, not about users.
You can find more information here : http://www.apple.com/itunes/affiliates/resources/documentation/itunes-store-web-service-search-api.html