Of course you can do that, but it is not the recommended way.
VLANs use software to emulate separate physical LANs. Each VLAN is thus a separate broadcast domain and a separate network.
As you have identified, routing between these VLANs would be difficult, because they are the same subnet. If the addresses are all different it is possible to route traffic using a very large number of rules which don't correspond to the actual subnet configuration and will confuse anyone who inherits this from you. However, it is completely permissible to use the same RFC1918 subnets on different physical networks. You could likely even make all the addresses the same.
The other constraint to bear in mind, and possibly the more relevant one, is that if any of these hosts have to connect to anything at all, routing them to that network will also be difficult. You would have to use NAT almost for sure, and set up NAT rules such that each of these VLANs has a separate outside address. If this configuration doesn't confuse the host OS, it will certainly confuse any administrator trying to work on it.
There are many, many, many RFC1918 addresses available, and there is rarely a real need to conserve addresses in this way. In the extremely unlikely case you are out of them, you can even use the RFC6598 address range
100.64.0.0/10 (which is designated as a private range for carrier-grade NAT, and though this is not its intended use, if you're large enough to use up an entire /8, /16, and /12 besides, you could likely make an argument that you are effectively the ISP for these devices).