The normal way? You can implement whatever kind of index you like — b-tree index, bitmap index, hash index are all useful and conceptually simple if you're familiar with the backing data structures.
For example, if you want to index a "foreign key" id stored in each "record" in a JSON array of objects, you build a hash table from the FK id values to the JSON array indices of the objects that have that id. It can be as stupid simple as an `fk_index = defaultdict(set)` somewhere in your program, to use a Pythonism.
Now when someone wants JSON objects in that array matching a given FK id, they can just O(1) look in the index to know the position of records that match. Much better than an O(N) scan of every item in the array.
Of course you have to to maintain the index as writes to the JSON happen, but that's not bad once you understand how things work. No real secret sauce.