I think the comments to this post set out the BSA position well and the typical way Scout Troops operate.
Any Scout Registered Scout can take any merit badge course they wish, (except for some defined requirements), as a Leader you should be encouraging the merit badges and guiding the Scout - obviously if one of the merit badges has an age restriction on it then that is different. The key as everyone has noted is that these are merit badge required to complete the Star, Life, and Eagle rank requirements, rather than Eagle Level merit badges.
As a Troop we try to sign up our new Scouts for the Swimming Merit Badge and the First Aid Merit Badge at Summer Camp (both are classed as one of the Eagle required Merit badges) . They can not get the First Aid Merit badge without the knowledge of the first Aid requirements for Second Class and First Class but they don’t need to hold the First Class Rank to complete the Merit badge.
A Scout could go through the program from 11 to 18 an earn merit badges only and never receive the Rank award of Scout. However that is unlikely as the Scout Award is relatively simple and should be pushed by the SPL, ASPL, Troop Guides for the new Scouts etc, and if they came from a reasonable Cub Pack with an AOL Award, should know most of it anyway. Tenderfoot to First Class bit harder to get and as a troop we have several kids who enjoy the trips, summer camp, etc but are not that interested in Ranking up.
These are difficult times and really you probably should talk to the parents to encourage the Scout to review some of the rank award stuff as well the Merit Badges and set up a SM/ASM sign-off session with him. Set him up to succeed not to fail.
Turning round and saying no to Merit Badges will result in you losing a Scout who appears to be interested in learning and moving forward with Scouts, just not quite in the way you want him/her to.