The 22-year-old has been linked with the Tigers for several weeks, but has now finally sealed his move to East Yorkshire where he will battle it out for the number one slot with Matt Ingram.
Baxter was also the subject of interest from League One side Sunderland.
The keeper spent last season on loan in League One at Accrington Stanley, making 19 appearances in all competitions and appearing in Stanley's 2-0 home win over Hull last term.
The Westminster-born player spent previous temporary spells at Ross County and Yeovil after earlier non-league stints earlier in his career with Met Police, Solihull Moors and Woking.
Baxter was signed by Chelsea as an eight-year-old while playing for Bromley side Cray Wanderers, and has worked his way through the ranks at Stamford Bridge and played for their academy sides.
The keeper is the Tigers' fourth signing of the summer following the earlier additions of Andy Cannon, George Moncur and Randell Williams.