Yes, the octal buffer needs a 5V supply. I would expect you to use the same supply that you use for your relay board. I would be very apprehensive about using USB power. If you are using an ESP32 development board that requires a 5V supply, I would recommend using the same supply. When using separate supplies there can be issues with parasitic powering when one supply is on and the other is off (see below). In this regard, don't overlook supply from a USB.
The buffer simply goes between the outputs of your ESP32 and the inputs of your relay board. The
datasheet shows the pin connections of the HCT244. Pin 10 needs to be connected to ground (0 volts) and pin 20 to Vcc (5 volts). Also pins 1 and 19 need to be connected to ground to enable the outputs. An output from your ESP32 would go to buffer input pin 2 ("1A1") and the corresponding buffer output from pin 18 ("1Y1") would go to an input of your relay board. However, I would recommend placing resistors in series with the outputs from your ESP32 to protect your ESP32, especially if parasitic powering could occur. The input current of each buffer is less than 1µA, so quite high values of resistor could be used but I would suggest something in the range 1kΩ to 10kΩ. (This aspect if often overlooked by electronic engineers.)
The datasheet says the absolute maximum output current (Io) can be up to "±35mA" (may be different for other manufacturers of this integrated circuit). That means the output can "source" or "sink" up to 35mA. So when the output is high it can source 35mA and when the output is low it can sink 35mA. That's good news because I don't know what the relay board requires: it should work regardless. However the 70mA limit on the total current is very disappointing, especially by modern standards (this is an old design of integrated circuit).
As this is an "HCT" integrated circuit, it is said to be "TTL compatible": the datasheet says the minimum input level that the IC regards as high is 2.0V. A high output from your ESP32 on 3.3V supply will easily be over 2.0V so that's OK. Don't get an "HC" version of this IC because its input voltage for high could be over 3.3V.
If you require a circuit diagram for discrete transistors, I will need to know whether the inputs of the relay board need to be connected to ground or to 5V to operate the relays (BTW: I've not worked out what the jumper on the board does).
If you are considering manufacture hundreds or thousands of these, we ought to consider other options more carefully
.