GRAMMER HAVE NOT..

Today we have not taken any photo yet.

This sentence is in present perfect tense. (have + past participle)
We use 'any' in negative sentences.
We use 'yet' in negative perfect tense sentences.

The correct format of present perfect tense is have/has + past participle. We use present perfect tense to describe finished events in unfinished time period. Therefore, sentence number 2, 'Today we have not taken a photo.' is the correct anser. It means today is not finished yet, and up till the time of speaking, you still have not taken a photo.