I've seen it done where a frame for the door was made out of 2x4s and the fence was stapled to the frame. Door mounted on frame. Keep the bottom strand of the fence intact so that you maintain the tension in the fence.
We did all our doggie doors with angle iron, welded onto the chain link fencing. Then we use an old mudflap, or piece of rigid plastic, just tied to the frame as the moveable door. Works well, but sometimes smart alpacas can get through.
What worked for my dog was to hang the interior gates a higher than you would with an exterior gate. The dog can just slip under them. I preferred that than reducing the integrity of the fence and also having to worry about any random pokey pieces of wire.
Be careful about hanging them high. I have a full grown male alpaca that loves to sneak under gates. I have some horse panels outside my compound when I let the boys graze in a section of my back yard. In about an hour I find him tromping around the unfenced portion of my yard wondering why he's not with the herd. I guess he thinks the grass is always greener on the other side!