I felt he had at last _____ his ambition.
Of the four options, I would say the following:
if someone or something reaches a particular point in their development, they get to that point
I'd not choose "reached" for this sentence at all. You "reach" a conclusion or a destination but it lacks the sense of accomplishment that needs to be included in saying that you've completed a goal. It's too bland.
to succeed in doing something, especially after trying very hard
This works for the reasons that "reached" does not. It shows that you've accomplished something... because that's the word. This is fine.
Obtain can often be simply replaced with "get", or in this case, "gotten".
to get something that you want, especially through your own effort, skill, or work [= get]
- Further information can be obtained [gotten] from head office.
- You will need to obtain [get] permission from the principal.
- the results obtained [gotten] through these surveys
You wouldn't "get" an ambition, so this option is out.
to successfully complete something or get a good result, especially by working hard
As you can see, this definition is very similar to "accomplish". Which is correct, as the two words (in this usage) can be treated as synonyms. Personally, I would prefer "achieved" in this sentence.
To summarize, "achieved" is the top choice for me, followed by "accomplished". The other two aren't options that I would use at all.
Other options...
There are tons of other options of varying popularity. I think that "accomplished" and "achieved" are probably better options in the end, though.
"succeeded in"
I felt he had at last succeeded in his ambition to play the violin at Kennedy Center.
"fulfilled"
I felt he had at last fulfilled his ambition to eat an entire pizza.
"attained"
I felt he had at last attained his ambition to lose 50 pounds.