Train Travel from Florence to Innsbruck

The train between Verona or Bologna and Innsbruck is run jointly by DB and OeBB and you can buy the ticket from one of them, while the section between Florence and Bologna or Verona is run by Trenitalia, so you will likely need two separate tickets bought from two different subjects.

There are two problems: first, while I remember seeing a DB ticket office in Verona, I do not remember such an office in Florence and I am not sure that Trenitalia will sell you a DB ticket for what they consider a competing service (you may try, however). Second, if your connection is in Bologna you can leave from Firenze SMN station, but there is only one such connection per day. All other connections leave from Verona, this means that you have to leave from Firenze Campo di Marte - as all services from Florence to Verona and beyond leave from this minor station.

Buying online the day before should not be a problem.

Trenitalia will happily sell you a through ticket to Innsbruck. There are several Eurocity trains a day from Verona or Bologna or Padua. They all have direct connections with Freccia trains from Florence, although the morning train leaves from Florence Campo di Marti station, which can be reached by local train from Florence S.M.N. or by taxi or bus from elsewhere in Florence.

Thank you both for the great advice!