Suppose we a have an anode and cathode and between them there is vacuum. If we apply a sufficiently high voltage between them, can we have a small electric current?
Yes, if you apply a high enough voltage, electrons will be able to leave the cathode (the potential energy from applied voltage need to be above the workfunction of the cathode material). Essentially you then create an electron beam. This is the basic principle of a vacuum tube, a cathode ray tube (tv tube) and an electron microscope.
It is even possible to create an arc in "vacuum". This happens when the cathode or anode material evaporates (every material, including metals, has a non-zero vapor pressure) and the gas (evaporated metal) then becomes ionized and becomes conductibe. Then of course we do not have a true vacuum anymore. The message is that as long as we have electrodes in a fixed compartment we cannot have ideal vaccuum. This in turn then can even lead to arcing. This for instance happens in vaccuum power switches.
Electric current is just the flow of electric charge due to the movement of charge carriers from one place to another. You don’t even need a cathode/anode or any source of p.d to constitute a current. If you set an electron, proton or any other charged particle in motion in an isolated vacuum, it will continue moving with a uniform velocity ( Newton’s first law) and thereby constitute a current.