“Je me fais toujours piquer” is how you properly say it.

Je me fais toujours piquer is the correct way to say it.

Toujours, just like any other adverb in French, is to be placed after the conjugated/infinitive verb or in the beginning of the clause (proposition).

Examples:

Usain Bolt court vite. [Usain Bolt runs fast]

J’aime généralement parler aux gens. [I usually like talking to people]

Toujours voir avant d’y croire. [Always see it before believing it]

Je suis évidemment prêt à le faire. [I’m obviously ready to do it.]

Le plus courant avec toujours:

Je me fais toujours piquer.

Egalement possible:

Je me fait piquer toujours.

Les adverbes peuvent se placer à peu près n’importe où, mais cela peut dépendre de leur rôle (associé au verbe, à un adjectif, à un groupe nominal).

Voir sur wikipedia des exemples et règles assez détaillées.

If the rest of the sentence is a condition, like “I always get stung when I sleep with my window open”, you would say “je me fais toujours piquer quand je dors la fenêtre ouverte”.

With less context, “tout le temps” would be much more common than “toujours”, although slightly more familiar : “je me fais piquer tout le temps” or “je me fais tout le temps piquer”. The difference in putting it before or after the verb is quite subtle : after the verb, the emphasis is on the temporal frequency ; before the verb, it’s more a relative frequency : “all the time” versus “every time”. It’s a question of context though, not every sentence will have that nuance, and it’s not valid for other adverbial locutions.

Adverbs are, in the vast majority of cases, just after the verb they modify. Here, even though the meaning is born by “piquer”, the conjugated verb is “fais”. So, “je me fais piquer toujours” is wrong : the adverb is too far from its verb.

One additional note, slightly off-topic : many people understand “je me suis fait XXX” as “I had someone / something XXX me” : I asked for it. “J’ai été piqué” doesn’t have that ambiguity : it’s something that happened, without any incentive from you. I’m not saying you should use the passive form : “je me suis fait piquer” is much more natural in colloquial speech. However, for sensitive subjects, it’s more careful to remove the ambiguity.