In your sentence as is 'her' is correct. In general 'herself' or any other reflexive is only used when the verb acts on the subject. For example, "I wash my hair." because the hair is receiving the action of washing. "I wash myself." because the subject 'I' is also receiving the action of washing. It isn't always that simple but that the general rule. I agree that what Greg wrote also sounds good but there are several correct ways to say most things in English, what you wrote is also correct if you use 'her'.

Actually, would sound better to say "Julia takes her dog with her everywhere. The order matters and you would not place the pronoun at the end in this case. Hope this is helpful.

here, her is correct because her is a possessive adjective which show possession or ownership of the dog . while reflexive pronoun would use only when subject and object of the same clause are same and it refers back to the same subject .