Yes, but this won't be by a conditional format. Conditional formats only alter the appearance of the formatted layout object not whether or not the user can interact with it.
but it seems to only apply to the current field and not related fields.
This is not necessarily the case, but what do you mean by "related fields". "Related" is a term with a very specific meaning when you talk in terms of a relational database. Do mean that the fields to which you want to control access are from a related table?
Would I need to set up a table and link the controlling field with it in order to make this work? Or is there some kind of more broadly useable conditional formatting strategy that will reference other cells.
I'm trying to understand your current design. There are many ways to control access to groups of fields. The best approach depends on how you design your database and exactly what you want to see happen. There are a number of ways, for an example, to make whole groups of layout objects appear and disappear from your layout. You can also leave them visible on the layout, give them a disabled appearance when access is not permitted and then use script triggers and scripts to prevent user access.
Using relationships linking to other tables or even the same table is one of the methods that can be used to do this.