It depends entirely on the grade, so a blanket answer would be misleading. 625 and 600 weld readily with matching filler. 718 is weldable but needs solution treatment and ageing afterwards to restore its properties. Cast gamma-prime alloys such as 713C are effectively not weldable, and that is a property of the alloy rather than a limitation of the shop.
The easy ones
Solid-solution nickel-chromium grades are the friendliest of the family. 600 welds with ERNiCr-3, 625 with ERNiCrMo-3, and both tolerate the heat input of ordinary GTAW or GMAW work provided the joint is clean and dry.
Cleanliness is not a detail here. Sulphur, lead, zinc and chlorine on the surface or on the filler cause cracking in nickel alloys far more readily than in carbon steel, so grinding dust, marking paint and chlorinated solvents all have to be kept off the joint.
The ones with a procedure
718 is weldable, and it is welded in aerospace and cryogenic service all the time — but the weldment is not finished when the arc stops. The ageing treatment that gives 718 its strength also promotes niobium segregation and laves phase in the weld zone, so the component usually needs solution treatment and ageing after welding to restore ductility and strength.
The C-family grades weld with matching filler, but they are sluggish and need controlled heat input, low interpass temperature and a clean, dry shield. High heat input coarsens the structure and costs corrosion resistance in the heat-affected zone.
The ones that are not weldable
713C and its relatives carry 5.5-6.5 % aluminium and 0.5-1.0 % titanium, which forms a high volume fraction of gamma-prime. That same precipitate makes the alloy strong at temperature and makes it crack when it is welded or hot worked. It is cast to shape instead, and repair is normally by re-casting or by a specialised process rather than by fusion welding.
If your design assumes welding a gamma-prime cast alloy, that assumption needs to change at the design stage, not in the shop.