A brand new rocket to be launched will have to be assembled, and that's a long process, though I do not know how long.
But if it's for an emergency, you may find ready rockets.
After the Columbia disaster, space shuttle missions all had a contingency mission in case they found issues with the orbiter before reentry.
The planning and training processes for a rescue flight would allow NASA to launch the mission within a period of 40 days of its being called up. During that time the damaged (or disabled) shuttle's crew would have to take refuge on the International Space Station.
You even had two shuttles on pad at the same time for a Hubble servicing mission, as the crew wouldn't be able to reach the space station for a safe haven, and would need to be rescued before the three weeks of consumables on board were exhausted.
The rescue mission would have been launched only three days after call-up.
That said the space shuttle isn't operating anymore, and until Spacex' Crew Dragon or Boeing's Starliner are operational (probably sometime soon), the Russian Soyuz is the only option to send people, and it seems to launch a new crew to the ISS roughly every 6 months (if there are no emergencies).
For the last flight of the space shuttle, the was no contingency mission prepared, but instead the mission had only 4 astronauts, and in the case that Atlantis couldn't make the reentry, the astronauts would stay on board the ISS and come back to earth in Soyuz capsules over the following year.
If your emergency doesn't require people to go to the ISS to help, but specific, unplanned cargo is still needed, a spacecraft (cargo or not) goes to the ISS every month or so, so if your emergency can wait between a week and a bit more than a month or so (to account for weather delays, mounting time of the payload and orbit maneuvers) you can just sneak it in one of those resupply missions, or replace some less vital cargo.
If your emergency can't wait at all, and you absolutely need to send something to the ISS on short notice, and there are no upcoming resupply missions, you may be able to hijack another vehicle with sufficient authority.
For March alone there are 10 planned rocket launches from the US, ESA or Roscosmos, so you could probably get emergency cargo to the ISS in around a bit more than a week (this is a guess, assuming the orbital maneuvers take around 3 days, and mounting the payload and waiting for a launch window take a few days too).
Note that if you don't have a proper cargo spacecraft to put the cargo into, maneuvering up to the ISS might prove to be difficult, and the cargo will not be able to mate with the ISS, so astronauts may have to perform an EVA to secure the cargo outside.