Upvote:2
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.
Upvote:2
I'm actually writing this from Keflavik airport after a two-day stopover in Iceland so here is some up-to-date (if unasked) points
I am looking to check-in to the hotel and then explore the city on the first day.
Good choice. Reykjavik is very walkable (provided you are dressed for the weather)
I am seeking some recommendations (name, address, website) for reputable tour operators, with physical store fronts near the city center so I can book some day trips during my 5-day stay.
SE explicitly bans specific product recommendation. IMO your best shot to look at TripAdvisor and Airbnb experiences, and see what's available . The reviews tend to be very reflective of the quality of the operator and you can contact them online, by phone or just walk there if you they list a physical address.