United Airlines operates out of all 3 terminals at EWR - terminals A, B and C - although the majority of flights arrive and depart out of Terminal C, and it's highly likely that both of your flights will be out of that terminal.
All 3 terminals are connected air-side by a shuttle bus. When using this bus, there is no need to re-clear security as it is air-side in all terminals.
All 3 terminals are also connected by an airtrain. This airtrain is outside of security, and thus if you catch it you will need to re-clear security. For this reason, if you do need to change terminals then taking the bus is recommended.
Looking at your specific flights today as an example, the flight from AVL landed at gate C109, whilst the flight to Dublin left from C82. These are both in the same terminal (C), however they are a fair distance apart.
To get between gates in terminal C (including gate that the bus goes to/from if you need to go between terminals) there are 'golf' carts available that will take you to your gate - simple flag one down, or ask the staff at the gate and they will contact one for you.
You can also contact the airline in advance and request a wheelchair, but unless you have severe difficulty walking this will likely end up being more of an annoyance than just using the carts.