New Hampshire law says yes:
661:5 United States Senator. If a vacancy occurs in the office of United States senator, the governor shall fill the vacancy by temporary appointment until the next state general election, when a senator shall be elected for the unexpired term.
That law relies on the authority granted the state legislature by the 17th Amendment of the US Constitution:
When vacancies happen in the representation of any state in the Senate, the executive authority of such state shall issue writs of election to fill such vacancies: Provided, that the legislature of any state may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
That's pretty clear, right? The legislature can empower "the executive" to make a temporary appointment. And who is the executive? Why, the Governor, of...
Uh oh... Wait a minute...
New Hampshire has this other part of state government: the "Executive Council." (Executive - there's that word again.) It doesn't pass laws. It's only function is to check the authority of the Governor.
The Constitution of New Hampshire spells out that role:
[Art.] 47. [Governor and Council Have Negative on Each Other.] The governor and council shall have a negative on each other, both in the nominations and appointments. Every nomination and appointment shall be signed by the governor and council, and every negative shall be also signed by the governor or council who made the same.
September 5, 1792
The federal Constitution allows the state legislature to empower "the executive" (not the 'Governor') to make an appointment to a US Senate vacancy. But the New Hampshire legislation empowers just one head of our Hydra executive, and fails to require the concurrence of the Executive Council.
That doesn't seem to meet the requirements of either Constitution.