The Idaho Constitution Requires Only That The Governor Do Whatever The Legislature Tells Them To
The Idaho Republican Party is making a sales pitch. The Idaho Constitution does not limit the role the Idaho state government is able to play - because it explicitly grants it a broad permission to add to its slate of responsibilities.
Article IV, Section 1: (Emphasis added)
EXECUTIVE OFFICERS LISTED — TERM OF OFFICE — PLACE OF RESIDENCE — DUTIES. The executive department shall consist of a governor, lieutenant governor, secretary of state, state controller, state treasurer, attorney general and superintendent of public instruction, each of whom shall hold his office for four years beginning on the first Monday in January next after his election, commencing with those elected in the year 1946, except as otherwise provided in this Constitution. The officers of the executive department shall, during their terms of office, reside within the state. Their official office shall be located in the county where the seat of government is located, there they shall keep the public records, books and papers. They shall perform such duties as are prescribed by this Constitution and as may be prescribed by law, provided that the state controller shall not perform any post-audit functions.
"As may be prescribed by law" means the Idaho Legislature, as a matter of constitutional power, is able to declare that the Executive has certain duties (such as, for example, state policing). Any such duty assigned to the Executive inherits its constitutionality from the passage of the law. This is further supported by Section 20:
Temporary agencies may be established by law and need not be allocated within a department...
Section 20 limits the number of departments within the Executive branch to no more than 20 and requires all duties and responsibilities to be re-organized thusly. This penultimate sentence in that section explicitly carves out the power of the legislature to decide to ignore this restriction (on an ad-hoc, temporary basis) when needed, and does not limit or constrain the purposes for which those agencies may be established.
Taken together, and absent any explicit constraint thereupon, there is virtually nothing the Idaho state government currently does which is not provided for by the state's constitution. That provision is, in fact, a requirement - specifically a requirement placed upon the Executive branch (Governor et al.) to carry out whatever the Legislature is able to pass.
The Republican Party's pitch here is flexible in the extreme, may be construed however they wish it to be, but allows them to drape themselves in the positive connotations held for such documents as constitutions. They are guaranteed to be in compliance with such a campaign promise, so long as they control a majority of the state's legislature and thus may pass or repeal whatever they please.