Observations from Earth indicate that at the solar surface, the outward magnetic field is the strongest at the polar regions.
Is at the solar surface an apposition which the writer has forgotten to add another comma (,) after that?
Nope, there is no apposition in the sentence, and there is no additional comma needed.
The sentence should be understood as follows:
Observations from Earth indicate the following: of all parts of solar surface, it is at the polar regions where the outward magnetic field is the strongest.
It doesn't make sense if "the outward magnetic field" is an apposition to "solar surface", since the main sentence would be
*Observations from Earth indicate that at the solar surface is [the] strongest at the polar regions
If you divide the above sentence into two parts you get (simplified):
a) Observations from Earth indicate something. What? b) That the magnetic field is strongest at the poles.
You see that in part b) "the magnetic field is subject. "at the solar surface" is a where-indication, as part of the that-clause it is an adverbial part. There should be two comas: ... that, at the solar surface, the outward magnetic field is strongest at the poles.
The best method to understand the structure of a sentence is to eliminate unnecessary elements. Then you get: Observations indicate that the magnetic field is strongest at the poles. Then it is easier to see how this sentence is built. Observations, subject - indicate, verbal part - that clause as object. In the that-clause: the magnetic field, subject verbal part: is + to be-complement (strongest at the poles)
Then you ask what is the function of the eliminated parts. from the earth: you ask: observations from where? This part is attached to "observations". at the solar surface: you ask where or what magnetic field. This where-indication is an additional element of the that-clause and called adverbial /or adverbial part of the sentence/clause.