What kind of "remote" is this? Many remote controls with a keypad have the switches wired as a matrix, and a solid state relay might not work.
A mechanical relay will exactly duplicate a pair of switch contacts. Go ahead and try the solid state relay, but if it doesn't work you may need a regular mechanical relay.
One thing I notice is that it says LED Foward Current = 50mA, Reverse Voltage = 5V, Peak Foward Current = 1A.
I am not sure exactly what those specs mean, and I don't want to order something only to fry it by using too high of a voltage(since it is an LED after all...)
Those are the "absolute maximum ratings", NOT the normal operating conditions!
With LEDs, you control/limit the current with a resistor. The supplied voltage gets divided between the series resistor & LED ([u]Kirchhoff's Law[/u]). Once the LED turns-on, the voltage across the LED is (approximately) constant, and the remainder of the voltage gets dropped across the resistor.
The specs say the LED will operate (turn-on the relay) at 3mA (minimum) and 1.25V. Let's say we are going to run it at 10mA (safely between 3mA and 50mA). With 5V applied and 1.25V across the LED, we'll have 3.75V across the series resistor. Using [u]Ohm's Law[/u], we calculate our resistor value as 375 Ohms. Since we have some leeway (we don't need exactly 10 mA) we can use anything in that ballpark. (330 and 390 Ohms are standard values.)