I'll start by analyzing the phrase:
a chance at that dream
Per the answer to this other question, the phrase a chance at [X] means an opportunity to achieve [X]. This fits with one of the definitions of chance: "a situation favoring some purpose; opportunity".1 Dream in this context means "a strongly desired goal or purpose".2
So "a chance at that dream" translates to "an opportunity to achieve that goal".
Now let's look at that definition of at from Merriam-Webster that you weren't sure about:
(Entry 1 of 6)
2. —used as a function word to indicate the goal of an indicated or implied action or motion
(https://www.merriam-webster.com/dictionary/at)
We've already established that dream is a "goal". And while a chance or opportunity is not itself an action, it encourages a course of action or makes it possible; in other words, it implies an action. So the at in your example phrase is "used as a function word to indicate the goal of an...implied action".
In short, the definition fits.
1 https://www.merriam-webster.com/dictionary/chance, entry 1, definition 2
2 https://www.merriam-webster.com/dictionary/dream, definition 4a