Learn German: Das Radar zeigt den Tornado. Gleichwohl nähert sich der Sturmjäger. - The radar shows the tornado. Nevertheless, the storm chaser approaches.

Gleichwohl is an adverb meaning "nevertheless," not a coordinating conjunction like aber. Because it takes up the first position in the second sentence, the verb nähert must come right next to it to stay in the strict second position (the V2 rule).

Could I just use trotzdem instead of gleichwohl in this sentence?

Absolutely. Both words mean "nevertheless" and trigger the exact same word order. Gleichwohl is simply a more formal, elevated synonym that you will often encounter in literature, news, or academic writing.

Why is there a sich in the second sentence?

It belongs to the verb sich nähern (to approach). In German, this verb is always reflexive, so the sentence literally says the storm chaser "approaches himself."

Doesn't sich nähern usually require a dative object to show what is being approached?

Often yes, as in sich dem Tornado nähern. However, if the destination is completely obvious from the context, you can leave the object out. The first sentence already establishes exactly what the storm chaser is getting closer to.