To expand on what your queueing options are:
Unless you can design the problem of hitting this rate limit out of existence as @Hammerbot walks through, I would go with some implementation of queue. The solution can scale in complexity and robustness according to what loads you're facing and how many rate limited APIs you're dealing with.
Recommended
You use some library to take care of this for you. Node-rate-limiter looks promising. It still appears you would have to worry about how you handle your user interaction (make them wait, write to a db/cache-service and notify them later).
"Simplest case" - not recommended
You can implement a minimally functioning queue and back it with a database or cache. I've done this before and it was fine, initially. Just remember you'll run into needing to implement your own retry logic, will have to worry about things like queue starvation **. Basically, the caveats of rolling your own < insert thing whose implementation someone already worried about > should be taken into consideration.
**(e.g. your calls keep failing for some reason and all of a sudden your background process is endlessly retrying large numbers of failing queue work elements and your app runs out of memory).
Complex case:
You have a bunch of API calls that all get rate-limited and those calls are all made at volumes that make you start considering decoupling your architecture so that your user-facing app doesn't have to worry about handling this asynchronous background processing.
High-level architecture:
Your user-facing server pushes work units of different type onto different queues. Each of these queues corresponds to a differently rate-limited processing (e.g. 10 queries per hour, 1000 queries per day). You then have a "rate-limit service" that acts as a gate to consuming work units off the different queues. Horizontally distributed workers then only consume items from the queues if and only if the rate limit service says they can. The results of these workers could then be written to a database and you could have some background process to then notify your users of the result of the asynchronous work you had to perform.
Of course, in this case you're wading into a whole world of infrastructure concerns.
For further reading, you could use Lyft's rate-limiting service (which I think implements the token bucket algorithm to handle rate limiting). You could use Amazon's simple queueing service for the queues and Amazon lambda as the queue consumers.