If you go to a person for help or advice, or to ask them for something with urgency, you can be said to run to that person.
Ex: Unfortunately, I have learned that if I say no to a spending request, the people in my office will run to my supervisor to try to have my decision overturned.
Ex: The young children were advised that if they needed help or got lost during the field trip, they should run to the nearest park employee for assistance.
If you make a request, you ask for something or make a petition. The word request can take the form of a verb, as above, but it can also take the form of a noun. A request is something you have asked for, either in written or verbal form.
Ex: I am not asking you to maintain complete silence; I only request that you keep your voices down so that we may hear the instructions when they are read.
If you request the pleasure of someone's company, you are asking them to be present at an event or to accompany you somewhere. This is a phrase that you may see on a more formal invitation.
Ex: Jill and Dave request the pleasure of your company at a dinner party to celebrate their recent engagement.
Ex: I have to drive quite a distance to pick up my mother from the airport tomorrow. I'd like to request the pleasure of your company for the ride, if you are able to come with me.