iceland – What is a well-established, reputable day-trip tour company in Reykjavik?

I have done what you planned to do, back in September 2014.
There are/were at the time quite a few companies and at that time it was easy to book only on the day before or even the same day for a half day tour.

I did not notice any unreputable tour companies. And I looked for online reviews as well as word of mouth in the hostel I stayed.
And I did notice that I did have much more choice when booking there compared to booking online before I went (as I had checked out and decided not to do.)

As weather is rather changeable, you may indeed find it a better option to adjust your plans to the weather. Going to see a waterfall in very heavy rain is not fun.

As Travel Stack Exchange we do not do company recommendations. And I think that now, after COVID 19 the field may have changed.