"I am finished with the website for now".
First of all, there's nothing inherently bad or wrong with using passive voice--often times, it is the better way to construct a clause.
Now, with that aside.
Does your example use passive voice? . . . Superficially, it might appear so, to an automated grammar checker.
A traditional grammar's version of a passive (voice) sentence is a sentence that involves a construction whose lexical verb is a past-participle verb form and whose auxiliary verb is a form of the verb "BE" (or of the verb "GET", in some uses).
Your example has the expression "am finished". The word "am" is a form of the verb "BE"; and the word "finished" has the shape of a past-participle verb form. But that is the crux of the matter: Is the word "finished" in your example a past-participle verb or a past-participle adjective?
There are some grammatical diagnostic tools/tests that you can try out to come to your own decision on this. One diagnostic test that can often be useful is the below--Is the candidate version similar in meaning to an active voice version:
"I am finished with the website for now". -- [original version]
"Something finishes me, for now". -- [active version?]
Hmm, that doesn't seem to support the passive argument. Compare to: "Today, the neighbors finished the job"; "Today, the job was finished (by the neighbors)".
(Aside: There's also the prepositional passive: "George Washington slept in that bed"; "That bed was slept in by George Washington". But that doesn't seem to be relevant here.)
Another diagnostic test, which can sometimes be useful, to see if the candidate word is an adjective, is to try to see if the candidate word can be modified by "too" or "very":
- "I am very finished with the website for now".
Nah, that isn't convincing either. Compare to: "Tom is very tired".
There are other diagnostic tests that you can try (which might be in your favorite grammar books), but at the end of the day, you'll have to make the decision for yourself. (I have my own opinion.) Someone else may very well produce a convincing argument for you for one position or the other. Of course, your decision might also be influenced as to the grammar that you are comfortable with, and with how that grammar defines passive voice.