You are permitted to travel outside of the US while in H-1B status. Generally, you need a valid, unexpired H-1B entry visa in your valid, unexpired passport for re-entry into the US. If your visa has expired, or you do not have an H-1B visa in your passport for your current period of H-1B authorization, you must apply for an H-1B visa at a US embassy/consulate abroad before you can re-enter the US in H-1B status. You cannot apply for renewal of an H-1B visa stamp while in the US.
Refer to the website of the consulate where you plan to apply for more information on what you will need in order to apply for an H-1B visa. Each consulate is different and may have different requirements. In general, you should plan to bring your original H-1B approval notice, a copy of your H-1B petition, a current letter from your MUSC department (on department letterhead) stating that you are still in valid employment status, and copies of your 2 most recent paystubs.
You must notify CGH in advance prior to traveling outside of the US in H-1B status. If your return is delayed or you experience issues with obtaining a visa while abroad, please notify CGH so that we are aware of your situation. Each time that you reenter the US, you must provide copies of your visa and your I-94 entry record to CGH. You will be able to access your I-94 arrival/departure record online at Arrival/Departure Forms. You can scan and e-mail these documents to firstname.lastname@example.org upon your return. CGH will then review your documents to ensure that everything is correct and that you are in valid status.
Please be aware that any time you travel outside of the US, you do so at your own risk. Re-entry into the US is never guaranteed.
For more information on traveling in H-1B status, refer to the following document: H-1B Travel (PDF).