The Senate rules for impeachment date back to 1868 and have been in effect since that time. They specifically provide for the subpoenas of witnesses, going so far in Rule XXIV as to outline the specific language a subpoena must use — the “form of subpoena to be issued on the application of the managers of the impeachment, or of the party impeached, or of his counsel.”
As you can see, there is no “Senate vote” requirement whatsoever in the subpoena rule. A manager can seek it on his own.
The rules further empower the chief justice to enforce the subpoena rule. Rule V says: “The presiding officer shall have power to make and issue, by himself or by the Secretary of the Senate, all orders, mandates, writs, and precepts authorized by these rules, or by the Senate, and to make and enforce such other regulations and orders in the premises as the Senate may authorize or provide.” The presiding officer, under our Constitution, is the chief justice. As such, the chief justice, as presiding officer, has the “power to make and issue, by himself,” subpoenas