Yes - in fact, we encourage you to do just that! Though you will be unable to use your short code until the carrier review process has finished and your application has been approved, you can still built and test your short code program using a standard Twilio SMS-capable phone number.
Twilio short codes use the same Programmable SMS REST API as standard phone numbers (AKA "long codes"). Therefore, anything you build to be used with a standard number can be easily used with your short code once approved - you'll only need to update the "From" number.
To get started, you only need to purchase a Twilio phone number.