Feature wise, Type-C is a superset of Type-B here, so this is definitely doable.
USB2 case can be implemented with only one PCB, one Type-C receptacle and two resistors. USB3 case is trickier.
Two possibilities:
This is an USB3 Type-B: The adapter mezzanine board should wire respective VBUS and GND all together (both orientations, both connectors) have one distinct 5.1kohm to ground on each of the CCx pin of Type-C. Then you need to do proper swapping of SS lines depending on orientation. For this, you have to take one of the CC lines (directly) and use it for controlling a SS diff pair mux (See figure 4-3 of Type-C spec). USB2 D-/D+ lines may be swapped with the switch (some have provision for this) or may be directly tied together (D- from both orientations together, same for D+) close to receptacle (USB2 spec stub allowances tolerate this, see Type-C spec, Table 3-4, note 1).
This is an USB2 only Type-B. For power, and CC, and USB2 do the same as above. Other lines (SSx, SBU) can be left floating. With tie-together option above, this is a passive-only solution.
In either case, you do not need a Type-C CC Port controller IC because you do not need Power-Delivery. Plug orientation will be reflected on receptacle CC lines passively (one will be tied to ground by cable, other will be pulled by remote DFP through cable. See figure 4-5). In the worst case, you’ll need a comparator to adapt to Threshold voltage of switch control pin. (Note some switches have built in CC handling)
In both cases, relevant impedance matching of data pairs should be taken care of.