This would be a basic networking issue, and most likely, you'd have to deal with this at your router/switch, as that should take care of inter-VLAN routing.
Depending on how that's done, it might not be necessary to define additional virtual interfaces on your Pi-hole machine at all - have a look at how slowlearner succeeded setting up multiple VLANs in Extra DHCP Server Options - #5 by slowlearner.
(As a side note: Starting with Raspbian Jessie, network configuration has shifted from /network/interfaces to /etc/dhcpcd.conf. While definitions in the former still work, you probably should verify that the latter doesn't interfere or contradict with them. Ignore me if you're already aware of this.)