I followed the dog. Luckily, as if it had become tired, or (had) reached its desired spot, it stopped.
Do I need the second had? Why or why not?
I think you can only get away with omitting had if the objects of the verbs are exactly the same and there isn't a comma or too many words between the two verbs.
I don't think she'll stop until she has completed or given up. (has does not need to be repeated.)
He seemed like he had been frustrated or angry with the results. (had been does not need to repeated.)
Here's a contrived example with verbs that are irregular in the past participle form:
Two days ago, while James was around, she had given him the money or gone over to his house.
Two days ago, while James was around, she had gone over to his house or given him the money.
Needs to be had gone or went for the first sentence, and had given or gave in the second.
Repeating had is optional. Also, repeating as or it would be optional as well. The key is the second phrase has to fit as if it were connected to the first. Consider
as if it had become tired
as if it had reached it's desired spot.
In either case, any of the repeated phrase (as if it had) may be omitted.
So you even could have said,
"I followed the dog. Luckily, as if it had become tired, or as if it had reached its desired spot, it stopped."
On an unrealted note, the comma after tired is not required provided you omit the subject (it) from the second clause. Correct punctuation would be:
I followed the dog. Luckily, as if it had become tired or (had) reached its desired spot, it stopped.
In my example I left the comma which repeated the subject (it).