Here are the stated requirements to be eligible for an H-1B visa. Reproduced the relevant section here -
For you to qualify to accept a job offer in a specialty occupation you must meet one of the following criteria:
- Have completed a U.S. bachelor’s or higher degree required by the specific specialty occupation from an accredited college or university
- Hold a foreign degree that is the equivalent to a U.S. bachelor’s or higher degree in the specialty occupation
- Hold an unrestricted state license, registration, or certification which authorizes you to fully practice the specialty occupation and be engaged in that specialty in the state of intended employment
- Have education, training, or progressively responsible experience in the specialty that is equivalent to the completion of such a degree, and have recognition of expertise in the specialty through progressively responsible positions directly related to the specialty.
I'm not privy to your current course, the key here would be if you're issued a Bachelors degree after the 4th year. If so, then you may be eligible. However if not and if you will receive both degrees after completion, then you currently have no degrees after high school and are not eligible for a job that needs a H-1B.
Am I able to start the application for an H-1B now, or would I have to
wait until I get that Master’s?
I'm not sure if you understand how the H-1B applications work, you can't start an application now nor later. H-1B is a non-immigrant petition made by a US Employer to employ a foreign national. In simpler terms, you don't apply for a H-1B, you apply for a job with a US employer. The US employer petitions USCIS and if selected, you can then go to the US and work for that employer.
A couple more points that may be relevant
- If you have a job offer and a bachelors degree, then you are against the clock. In the last few years the H1 window opens during the first week of April only and is oversubscribed by 4:1. There is a lottery which determines the winners.
- The answer here assumes that you're going to be working for an employer who is not Cap Exempt (most employers like IT, Finance, etc)