Dwarven Fortitude should continue to function while in Wild Shape.
The description of the Wild Shape feature states, in part:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
So Wild Shape allows us to use feature from both our class and our race.
A feat having our race as a prerequisite should be viewed as something like "a feature from your class and race". Or to put it another way, meeting a racial prerequisite for a feat should be seen as a benefit or feature from your race, which carries over in Wild Shape (as long as your new form is capable of making use of that benefit).
Viewed this way, the relevant benefit of the Dwarven Fortitude feat should continue to function just fine. Or at the very least, the race prerequisite should not be an obstruction to using the feat while in beast form. (That said, it is unfortunately not as explicit as we often wish rules interactions were, so there is room for a DM to rule otherwise.)
As noted in this Q&A about the ability score increase from a different feat, you will not retain the increase to Constitution from the feat. However, the second benefit of the Dwarven Fortitude feat states (XGtE, p. 74):
Whenever you take the Dodge action in combat, you can spend one Hit Die to heal yourself. Roll the die, add your Constitution modifier, and regain a number of hit points equal to the total (minimum of 1).
Since your beast form is physically capable of taking the Dodge action, you should be just fine to use this feature of Dwarven Fortitude while in beast form.