No, you can’t use a subjunctive here. Subjunctive is used to express a possibility, a wish or a desire. Here you are staying a fact, so indicative mode is required.
Short answer: No, you’re fine with the indicative here.
Long Answer : Firstly, you only need* the subjunctive with a superlative when there’s a que in there. For example, as Simon pointed out in the comments
Il est le plus sportif que je connaisse
However, you might have noticed the star next to the need up there. That’s because the subjunctive following a superlative is optional, depending on what you want to say. If you used the subjunctive, you’re implying that he might be the most sporty, but you’re not sure. Since in this case, you know your family and you are pretty sure he is the sportiest, you want to stick with the indicative. (Some better examples can be found in the link I gave below.)
If you want to read a bit more about it, simply googling “Subjuncitve after superlative french” will get you a wealth of reading, but this website is always good for decent grammar explanations.