Most of the time, an orchestra will tune to the oboe.
Why is that? What is the next instrument if, for some reason, there are no oboes?
I found this:
Circumstances of history, mostly, but also acoustics. The first orchestras (in the late 1600s) were mainly string instruments. A pair of oboes was sometimes used to strengthen the first and second violin parts. Soon composers were writing separate parts for the oboe, exploiting its singing tone as a contrast to the violins. The bright, rather penetrating sound of the oboe was easy to hear, and its pitch was more stable than gut strings, so it was natural to rely on it for tuning (One can also imagine it settling, or preventing arguments. Twenty string players squabbling over a tuning note, then asking the oboist to intervene). Other instruments drifted in and out of the orchestra – flutes, bassoon, French horns, clarinets – before it’s instrumentation became relatively standardized as we know it today. But oboes were almost always present, so they became the standard instrument for tuning.
And reading the link provided by @leftaroundabout in Why tune to the oboe? we can say the reason there is mostly because of historial reasons. In other words, it is a habit.
An oboe is an instrument with quite rigid pitch (possibly somewhat depending on the reed, with oboe reeds having pretty much the shortest lifetime of all reed instruments). String instruments can be painlessly retuned, and most wind instruments can be tuned a bit more or at least can be better pitch-shifted when playing using embouchure, air pressure or tilting.
You probably don't want to dick around with the basic pitch of an oboe reed that happens to work well at the moment. It's not likely to achieve much and may well shorten reed life. Also the default amount of lip and air pressure is already at the top of instrument scale (as a bit of compensation, the actual air flow is quite small, making oboe the prime contender for circular breathing techniques) so there is not likely a lot of variation possible without danger of leaving the sweet spot of amicable tone production.
So tuning to the oboe is in the best interest of the concert.
Because the oboe player can sound the A with only one hand. The left hand, of course, being busy holding a tuning fork to their ear.
I have not been able to find any online source for this, but I heard it from a normally trustworthy source. Makes a lot of sense to me. Of course, the matter of why we tune to the A4 to begin with is related...
Nowadays, the trumpet can equally easily play a concert A4 with one hand, but the trumpet would not always have been there in baroque times (as indicated by SysDragon's source).
Basically, You can't tune an oboe, but you can tune everything else, so at least the orchestra will be in tune with itself. No one in the audience will notice; everything is relative.
If you have ever seen an oboe on an oscilloscope, it produces a perfect sine wave. As contrast a trumpet's output has a much more complex sound wave. An experienced oboist can produce a perfect standing wave. Today we have electronic machines that can do this, but in the mid-17th century they had to use natural means. This has carried on through tradition till today, even though we can do this by machine now.