Generally not a good idea unless that other person contacts you. It doesn't hurt to follow up with Person 1 to ask if Person 2 has said anything, especially if it been more than 2 business days since your contact.
The reason being is that you're contacting a representative of the company who is taking your request and channeling it to the appropriate person. By contacting the person directly then you sort of lose the ability because that means that person has to forward your email back to his/her boss, then see if the task is okay to proceed. That person may also decide to ignore your request because you're contacting them directly and they do not have a task assigned to them to help you.
At a really good company, they would listen to their clients, price it, draft up a contract/agreement/pricing plan, then they would give it to their project managers who would then give it to the developer/designer/workers.Once the task is done it goes back to the client for evaluation.
Contacting the bottom to the top is never a good idea. My thought is wait, or ask the rep you're talking to who is the best point of contact. But I would just wait to hear back on your request then ask if you have any more questions.