The time of submission of your specimen depends on the type of application you file with the USPTO.
There are two types of applications - "intent-to-use" and "use in commerce".
If you are not commercially active in the US market when submitting your application, you must choose "intent-to-use" filing. In this case, you will be asked to provide proof of use (specimen) when your application passes the examination stage. It usually takes around a year from the submission of your application. You must pay an additional $100 per class at the time of specimen submission.
If you file a "use in commerce" application, you must attach proof that you are commercially active in the US market. This means that your goods and services can be accessed/purchased by your US customers. Shipping to the US is sufficient; your company does not have to be based there. There are no additional fees when opting for the "use in commerce" filing.