Does a radio antenna need to form a closed loop to function?
An antenna of the classical type (such as a quarter wave monopole) appears to have an open end and therefore, current flowing in the monopole must become zero at that open end. This kind of implies that no current can flow but that isn't the case. A monopole requires the presence of ground/earth to work effectively and, it is the capacitance to ground that allows current to flow in a seemingly open-wire. Of course that current has to be AC just as a capacitor requires an AC voltage across it for current to pass through it.
It also needs to be roughly the right sort of frequency compared to the physical size of the antenna to transmit or receive radio waves effectively.
Yet real current (electrons) do not pass from one capacitor plate to the other; the action of charge building up on one plate ejects charge from the other plate and current is &quot;seen&quot; to flow through the wires of a capacitor. What flows between the plates is called &quot;displacement current&quot;. Displacement current has all the same hallmarks as a real flow of electrons without electrons being harmed in any way. That displacement current produces a magnetic field just as a real flow of current does.
So, when we have a monopole excited with an AC source, there are magnetic and electric fields produced that surround the antenna. Those fields can be quite complex close to the antenna but, at some small distance from the antenna can resolve into values that match the impedance of free space (approximately 377 Ω). That creates a radio (EM) wave that propagates at the speed of light. In fact light is an electromagnetic wave and it propagates through space using the same medium as radio.
A dipole antenna is two quarter wave monopoles and doesn't require an earth/ground because it is driven differentially and, it is the capacitance from one end to the other that is important: -

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And, when receiving: -

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With both dipole and monopole antennas, if you match the applied frequency to the length of the antenna you can efficiently convert electrical power in the feed wires to radiated power in free-space (vacuum or air for instance).
See also Why does 1/4 wavelength have a ground plane and 1/2 wavelength needs none?.