Learn Portuguese (Portugal): Com este nevoeiro, não vejo a rua.

In Portuguese, you often do not need to use the verb poder (to be able to) with verbs of the senses like ver (to see) or ouvir (to hear). While you could literally say não posso ver, simply saying não vejo (I do not see) naturally expresses 'I cannot see' in this situation.

Why do we use este here instead of esse?

The word este (this) is used for things that are physically present with the speaker. Since the speaker is currently surrounded by the fog, este nevoeiro is the correct choice. If they were looking at fog far away, they would use aquele, and if the fog was near the person they were talking to, they would use esse.

Is vejo an irregular verb?

Yes, vejo is the 'I' form (first-person singular) of the irregular verb ver (to see) in the present tense. Most regular verbs ending in -er simply take an -o in this form, but ver changes its stem to become vejo.

I know nevoeiro means fog, but I have also heard the word névoa. Are they interchangeable?

They are related, but nevoeiro specifically means thick, heavy fog that severely blocks your vision, which is exactly why the speaker cannot see the street here. Névoa is used for a lighter mist or haze.