So in Paris, is there a particular way to request a taste of cheese? Or does one just wait to be offered?

This is important since when I asked my 7 year old what type of museum she wants to visit in Paris, she said, "the cheese museum!" I think we'll be tasking a buying quite a few petit morceaux du fromage.

S'il vous plait, puis-je le déguster? (Please, may I taste it?)
Pourrions-nous le déguster? (Please, may we taste it?)

That's maybe the most polite way to say it? The ordinary question structure, "Est-ce que" would work, too.

"S'il vous-plait, est-ce que je le déguste?" "S'il vous plait, est-ce que nous le dégustons?"

I find that prefacing my questions with "S'il vous plait" (please) works very well in terms of getting people's attention and establishing a pleasant interaction.

But I don't think I've ever had to ask to taste the cheese. Demonstrate an interest and they'll probably offer.

Don't forget to greet the shopkeeper/stallkeeper ("Bonjour madame/monsieur"), and wait your turn (which can take a while).

Some words you might hear the shopkeeper use when proposing different cheeses (my spelling will fail me on this, so please indulge me):

Doux/gouteux (mild/strong-tasting)
Moeulleux/sec (soft or runny/dry)
Vieux/jeune (old/young)

"Est-ce que je peux goûter (un petit morceau)? is what you would say, not "déguster".