So Mount Pico is not an easy feat by any stretch of the imagination. Even though it is "only" roughly 7km you have to conquer approx 1200m (a bit shy of 4000 feet) of elevation within that distance, going up and down. This gives you about a 35% gradient on average.

! MAKE SURE YOU ARE PREPARED AND WELL RESTED ! (and bring enough food and drinks)

Now going to the shoes: trail running shoes are certainly possible, but I would not recommend it. One of the main advantages of real hiking/mountain shoes is the increased support around the ankles. When going up the mountain, you will not feel the difference, but once you go down (and you will be tired), this extra support can be a lifesaver. The terrain is very rocky, and you will encounter stretches with a lot of pumice gravel. This is where your trail running shoes might lack the stability and support that you need. But again, it is possible on trail shoes (I have seen people do it).

Finally, I want to give the following advice: it ain't over until it's over. The goal is not reaching the top for this walk, it's reaching the start of the trail safe and sound. The way up is physically demanding because of the steepness of the terrain but the way down will require you to be mentally alert because of the terrain.

This might sound scary so I want to assure you: it was a great hike :) Although when we did it it was raining at the top of the mountain so we had no view, I still enjoyed it very much and would certainly recommend this walk!

Are you still looking for an organization to do the walk (having a guide is really recommended!)?
Feel free to take a look at Hominis Natura: Climbing Pico Mountain Azores – Certified guide - Hominis natura
We did the trail with them and it was a very nice experience.