H1-B Visas are Established by Law
The H1-B visa is allowed for by the Immigration and Nationality Act. The relevant text is (Sec 101 (H)(i)(b)):
[someone] coming temporarily to the United States to perform services ... in a specialty occupation described in section 214(i)(1) or as a fashion model ...
This portion of the text is the definitions. It establishes people in this category as being aliens who are eligible for non-immigrant visas.
Statute Does Not Require Them
However, the same statute defines only the maximum number of entrants to the country (Sec 203(b)(i):
Visas [for priority workers] shall first be made available in a number not to exceed 28.6 percent of [the total limit]
There is no requirement to issue any visas at all - only a maximum proportion that may be issued. It is conceivable that the President could direct his Department not to issue these visas for a given year. At the very least, this would eventually to all H1-B visas not being renewed and through attrition those people would leave.
Revoking Existing Visas
The executive branch's regulations allow the Department of State to revoke any visa for any reason:
A consular officer, the Secretary, or a Department official to whom the Secretary has delegated this authority is authorized to revoke a nonimmigrant visa at any time, in his or her discretion.
The President could direct the Secretary (and their consular officers) to revoke H1-B visas (which are non-immigrant visas) completely at their own discretion.