Synonym with positive connotation for "peeping through the door"

Solution 1:

The word 'peeped' suggests something surruptitions and illicit. It is strongly associated with "peeping Tom", someone who spies on people.

"peered" means roughly the same as "peeped", but without the secretive and negative connotations. "Peeked" might ulso be viable.

Solution 2:

Yes, peeped through the door suggests something secretive or cautious. That said, it is probably okay to say:

She peeped in asking for permission to enter.

Or, using a close cousin:

She peeked in asking for permission to enter.

The usual phrases also involving "the door" run along the lines of:

She stuck her head in the door asking for permission to enter.

She poked her head in the door asking for permission to enter.