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Due to a voltmeter’s strong resistance nature, how could electrons flow from anode to cathod? It is more reasonable to assume that a voltmeter would stop the redox reaction because the electrons could never reach the positive electrode. When I measure the potential energy between two electrodes with a voltmeter, is the reaction happening? If the reaction is not taking place, how come the potential difference exists?

An ideal voltmeter has an infinite resistance, so does not let any current flow in the system. However nothing is ideal in real life, it consumes a very small amount of current to move the magnet and needle arrangement or electronics. An ideal ammeter has zero resistance. Assume that you have a water tank on your ceiling which supplies water to your kitchen tap. If I add a pressure gauge before your closed kitchen tap, it will read a positive pressure, much greater than the atmospheric pressure. Note that there is no water flow. Forget about the textbook picture of measuring galvanic cell voltages using a voltmeter (as shown in chemistry textbooks), as nobody every measures cell potential this way. A reliable approach to measure cell potential is called null-point detection (you can search more about it). As the name indicates, it draws zero current at the point of balance condition. It is one of the most precise and accurate techniques in classical potentiometry.

Just think a little deeper, the two electrodes in a galvanic cell do not "know" each other and they are oblivious to each other's presence. So current flow is not a requirement to measure potential difference. Each single electrode is in contact with its ion in the solution. If you have a zinc and copper galvanic cell, each has its own electrode potential e.g. a Zn rod dips in Zn2+ soln and Cu rod dips in Cu2+ solution. Electrode potential is the thermodynamic tendency of ions (in solution) to reduce themselves without any current flow, in this case Zn2+ --> Zn and Cu2+ --> Cu. The voltmeter simply reads the difference of potential between zinc and copper, however it does not know the individual electrode potential of zinc and copper.

Yes, the reaction must be taking place for the voltmeter to measure the difference in potential. Remember

$$V = iR$$

thus a voltmeter works by knowing its internal resistance, $R$, and measuring $i$. Also notice that if $i=0$ then $V = 0$, so a voltmeter must have at least some tiny current to work.

The problem with a voltmeter is that you can imagine that a battery has an internal resistance. So the more current flowing through the battery the less voltage will be measured by the voltmeter. Thus the cell voltage measured by a voltmeter would be low, at least theoretically.

$$V_textmeasured = iR_textmeter - iR_textbattery$$

Modern voltmeters with integrated circuits and operation amplifiers have a very large internal resistance, and thus draw a very very small current since $R_textmeter gg R_textbattery$

In the not to distance past voltmeters drew too much current to give an accurate reading for a cell, so a Wheatstone bridge was used. But even though a Wheatstone bridge is used to detect "no" current, there is still some finite sensitivity, so the ideal of absolutely no current flow is certainly not reached. The other weakness of a Wheatstone bridge is the need for high precision resisters. So to measure the galvanic cell potential a Wheatstone bridge no longer has any practical usefulness.