I've tried to search for company register with API for registered businesses (including companies and self-employed people) in Austria, but with no luck. Does anybody know, if is there any?
There is a Firmenbuch in Austria which officially registers all businesses. Unfortunately, retrieving data from that register is not for free. Take a look here, and you will see a price list after your search query. (If you know German, there are more information about the Firmenbuch here at the Republic of Austria's Unternehmensserviceportal.)
If you are fine with aggregated data about the Austrian landscape of businesses, then you could take a look at the relevant page of Statistics Austria. You might also find something at Austria's Open Data Portal.