Why does Ohm's law not work for vacuum cleaners?
The 7.7 ohms you measured is the winding resistance of the motor. But that is not the only factor that determines its operating current.
Your vacuum cleaner might draw close to the calculated 30A the instant power is applied, but as soon as the motor starts to rotate, it generates a voltage that is proportional to speed (called back emf) that opposes the applied voltage, decreasing the net voltage available to drive current through the windings. As the motor speed increases, the current (and therefore the torque produced by the motor) decreases, and the speed settles at the point where the torque produced by the motor matches the torque required to drive the load at that speed.
Fuses don't blow instantly. But if you were to lock the motor so it couldn't rotate, that fuse wouldn't last long.
A vacuum cleaner isn't a resistor, and the line voltage from the outlet isn't DC (Direct Current). Ohm's law applies to resistors and DC. Ohm's law doesn't directly apply to your motor connected to an AC (Alternating Current) source.
For motors, you need to look into the rules for alternating current and inductors. They are far more applicable to your case.
"Resistance" is for DC circuits. While resistance still plays a role in AC, there is also another characteristic for AC circuits called "Reactance", which is effectively just resistance to alternating current. "Reactance" is provided by inductance and capacitance and changes with frequency, per the following formulas:
$$X_L = 2\pi f L$$ $$X_C = \frac{1}{2\pi f C}$$
where \$X_L\$ is inductive reactance (in ohms), \$X_C\$ is capacitive reactance (in ohms), \$f\$ is frequency (in Hertz), \$L\$ is inductance (in Henrys) and \$C\$ is capacitance (in Farads).
Together, resistance and reactance (whether inductive or capacitive) become a complex number of the form
$$Z = R \pm jX$$
where \$R\$ is the resistance, \$j\$ is an imaginary number (\$\sqrt{-1}\$), and \$X\$ is the reactance. The resulting complex number is called "impedance", denoted by the letter \$Z\$, which affects current draw of your device. You can use \$Z\$ in place of \$R\$ anywhere in Ohm's law and it will work, but you must do the math properly with the complex numbers. It is a bit more difficult, however, because there is a lot more to a motor than just inductance, for example. The windings themselves have capacitance and resistance, so it may be difficult to find all of the necessary variables in order to accurately calculate current.