As has been mentioned in a comment to the question, the adverb osmotically (see Merriam-Webster's definition of osmotic) is perfectly valid. So, if you are fine with the base word, then just use that.
If you want a different, single word, then my preference would be vicariously, also as mentioned in a comment.
From Merriam-Webster's definition of vicarious:
1 : experienced or realized through imaginative or sympathetic participation in the experience of another
// a vicarious thrill
So, your sentence could take one of these forms:
When my daughter's project won the science fair, I felt pride vicariously.
When my daughter's project won the science fair, I felt a vicarious pride.
Note that I removed the word too.
I am assuming that you mean to say that your daughter feels pride—and so do you. But if you say I too felt pride vicariously, that would mean you felt pride vicariously just as somebody else (your daughter) felt pride vicariously. However, your daughter's pride is not vicarious, it's direct. So, the too provides an unintended meaning.
Or you may actually have meant to use too but left out some other information:
As the other parents felt with their children, when my daughter's project won the science fair, I too felt pride vicariously.
Now it can be understood that you are joining the other parents in feeling pride vicariously.