It is never explicitly stated in the script, but it is rather strongly implied. Aside from Munson's obvious disdain for the spaceship, which would be harder to explain if Zarkov had stolen an intact rocket from NASA, we have the following. Note that the scene plays out the same way in the movie as it is written here, but the design of the rocket was apparently changed at some point between writing and filming.
[The rocket] is curiously transparent from the inside, mirrored only
on the exterior. There are some controls and displays, but
in general the interior is severely ascetic: as far removed
from conventional spacecraft as ZARKOV's solitary genius is
from everyday science.
Immediately starting to set switches in a frenzy of activity,
ZARKOV waves the gun at FLASH.
Sit down there! Foot on the red pedal!
Flash! Sack him!
ZARKOV wheels and pulls the door shut. FLASH jumps him. He
punches ZARKOV in the gut. ZARKOV slams him back with the
well-known strength of a madman. FLASH clobbers him again.
Falling, ZARKOV dives sidewards and hits a switch.
The capsule WHINES SOFTLY and begins to WHIRL. Looking out
through the transparent wall, it is as if the capsule itself
was motionless and the scene outside is whirling. What's
left of the greenhouse is under a constant rain of those
LASER BOLTS, flashing fire that makes us dizzy.
FLASH staggers to his feet and finds DALE, folds his arms
around her protectively. The centrifugal force is rapidly
increasing now, throws them both against the wall. Faster
and faster, whirling, the force presses DALE'S body back
against FLASH'S until they are almost one flesh.
We move over and find ZARKOV plastered against another part
of the whirling capsule, his face drained white and distorted
by the terrific G-Force operating here now, his lips moving
as he speaks with enormous effort:
Friendship...Built this to send to
them in...Friendship...The end
now...End of the world....Unless we
(every sound an effort NOW)
ZARKOV'S eyes close. He loses consciousness.
He says he "built this to send to [the aliens] in friendship", although the "I" is implied rather than explicitly stated. The script makes it fairly clear that the rocket is a totally unprecedented design, nothing like the rockets used by NASA, and suggests that this is due to "Zarkov's solitary genius". The clear implication is that Zarkov somehow built the rocket himself - with at least some help from Munson - to his own specifications. We don't know where he got the resources and materials to do so.