How can the order of a transfer function be derived from its equivalent state space representation?
One trick will be whether there are any pole-zero cancellations in the overall transfer function.  For that, you will probably need to check for the minimality of the system (i.e. check whether the system is both controllable and observable).

The question of "determine the order of the transfer function for each input" seems odd, to me.  If the system is minimal (it can't be represented by a smaller order system), then you need it as-is.  Pulling out one of the $m\times 1$ systems on its own may be of interest, but you can't reduce it without destroying the overall system.

Can you give more information about why you want to examine just one of the $m$ transfer functions?