Can a Arduino MEGA 2560 R3 provide enough current out of its pin to activate the Songle srd-24vdc-sl-c relay? Will powering the Arduino with an external power supply help?
This relay cannot be directly commanded by any Arduino because its coil require 24V voltage to be able to switch the contact; Arduino Mega pins output voltage is 5V.
Regarding current, the relay coil needs 15mA ("high sensitivity" model) or 18.7mA (standard model), that would be OK for an Arduino Mega pin (max current = 40mA).
Anyway, you would have to add some circuit to adapt voltage between the Arduino pin and the relay. For this, first of all you will need to get a second power supply (in addition to the Arduino 5V supply used for Arduino) that can supply 24V.
Then you could use a MOSFET transistor between the Arduino pin and the relay. A MOSFET is often used in so-called Solid State Relay (SSR) because it can almost do the same as a coil relay but is 100% electronic-based (no moving parts); however, contrarily to a coil relay, it does not provide galvanic isolation between the input and the output circuits.
You can check out this question to find out further information on MOSFET.
Personnally, I would rather replace that relay with a 5V relay model, as that would make the circuit simpler, and would remove the need for a second power supply.