The direct answer is no, it is not possible to compile a VB6 project with OCX dependencies without those dependencies being registered.
Furthermore, the act of compilation itself involves VB6 attempting to register what it has just built (unless you are compiling to an EXE). This generally requires the VB6 IDE and/or its compiler to run with "admin" permissions. Therefore the permissions are a hard to avoid issue regardless.
I believe these issues can be obfuscated by the fact that VB6 itself (the IDE and/or the runtime) will sometimes try to automatically register certain things for you, but will keep silent when it does so.
You should probably create a different process to setup a development PC from the build process you use from deployment. This may "feel" wrong especially if you have experience with other programming environments, but I would stress that VB6 can be very painful & problematic to work with and so pragmatism is generally in order.
On the development PCs: Setup all the unchanging dependencies once (and document them) and then leave them alone (as noted in another answer.) When weird dependency problems occur, verify the PC is setup correctly before doing anything else.
If you have all the sources to your dependencies, then I would consider if you can actually run them all in a VB6 project group (VBG) and not compile them at all. (A VBG is akin to a .NET solution though far less powerful.) I do this often and it cuts out a lot of wasted time. Developers don't necessarily need code compiled to EXE / DLL / OCX - they often just need to be able to run it in the IDE.
On the build PC: If you can always start with a clean environment, like in a virtual machine, then I think its actually a good idea to register everything from scratch in an automated fashion as this helps to verify nothing is missing or mismatched. Re-using the same build environment without doing this can mask problems when some dependency has changed in source control but still exists on the build machine. On a VM generally permissions aren't a limiting factor.
Notes:
If you are building an EXE, VB6 does not require any elevated permissions, as far as I can recall.
Running code in the VB6 IDE does not either.
[Caveat 1]:
It may technically be possible to create a side-by-side application manifest file for VB6.exe itself and include in that manifest whatever dependencies you need, thereby avoiding having to register them.
But this would fall well outside of the normal ways to use VB6 tools - its a hack - and possibly is not worth the potentially large effort. I don't think I've ever seen a working example and so I don't recommend this as a practical solution, but mention it for completeness.
Maybe in some locked-down corporate IT scenario this could pay off... maybe. In that scenario doing dev work in a VM might be a better option though.
