do you have a local petting farm near you??
i would start by reading her lots of animal stories about freindly sheep and pigs.
the arrange a trip to the farm ( they are usually either free or very cheap)
on the first visit let her do what she wants to do. if she is scared then carry her if thats what she wants.
then arrange another trip a week or so later.
this time dont carry her. dont force her to go anywhere near the animals, just insist she walks by herself..even if she is holding your had.
then again the following week (sorry but it really will need a couple of visits)
this time when your elder daughter is happily stroking or being near teh animals make a real fus of her. say" what a brave girl you are stroking the animals. arent they so freindly. they really like you being their freinds...that sort of thing.
it may nothappen on this visit but if you continue going every so often and you continue this pattern of praising your eldest daughter your younger dd will want to get some of that attention too. you will notice her slowley getting closer. dont encourage her but DO praise when she does move forward. so not things like " here DD come stroke the sheep" but things like "wow your standing so much closer arent you, your really seeing how nice these animals are. well done im very rpoud of you"
as i say it may take a couple of praise visits for her to build the courage to actually touch an animal but she will. i promise.
just remember baby steps and lots of praise.
you can of course just go to a local park where lots of dogs go for walks but you can never really predict which dogs are freindly. at least at petting farms/zoo's you know the animals are used to lots of child interaction.
good luck. hth