In theory you can run MPLS without an IGP, any sane network admin would not though. MPLS labels can be assigned through:
Generally a router runs IGP to populate the Routing Information Base (RIB). This is the control plane, routes are then installed into the Forwarding Information Base (FIB). This is the data plane forwarding, depending on platform, there may also be distributed forwarding on linecards so the linecard will have a local database.
When running LDP, labels are advertised to adjacent routers based on the RIB. The labels are then stored in the Label Information Base (LIB). This is the control plane and equivalent to the RIB but contains labels. The LIB is then one of the components used to build the Label Forwarding Information Base (LFIB) which is the equivalent of the FIB.
Running LDP, without an IGP, the labels would not be assigned because there are no routes in the RIB. This could be solved by entering static routes but it's obviously not a scalable solution.
When using RSVP-TE, labels are not assigned through the RIB. Tunnels are manually defined, connectivity is still needed between the headend and tailend, which is usually done through IGP but could be done through static routing as well.
There are special cases where routing is not needed with RSVP-TE because the neighbor is statically defined such as in the configuration below:
ip address 18.104.22.168 255.255.255.0
mpls traffic-eng tunnels
mpls traffic-eng passive-interface nbr-te-id 22.214.171.124 nbr-if-addr 126.96.36.199
Example above came from RSVP-TE without IGP
So the configuration is possible but definitely not recommended. To be able to reach the router remotely for management you would want to have an IGP running, also possible to solve through static routes of course. In the end it's about scalability and how prone to error each solution is.