How do I seal the gap in the door where a small section has rotted away for the least amount of money.
You do not say where the gap is located but usually, if the door is wood then the gap would be at the bottom.
They make specific size 'runners' for doors but you will need to know what size you need (width) and how much clearance you really have between the door and the floor.
There are many online sites that sell strips for this type of repair but Amazon has a large selection so you can look at these and see if there are any you think would work for you.
www.amazon.com/
You can also measure your door width and visit a hardware store to look at what they have available. I like using Ace and TrueValue hardware stores as the employees seem to take more time trying to help me. Almost all of the big box stores sell the strips and you can also just buy some of their felt for door bottoms and cut and glue what you need.
If you have gaps in other areas there is also strips for windows and door frames.
www.amazon.com/
If only a small area of rot damage, try using a good wood filler that you apply and smooth out with a putty knife. After it has hardened, you'll need to do some sanding to smooth the surface. Then apply primer, and when dry, apply at least two coats of paint or sealer to match the color or the door. If it is a gap across the whole width of the door because of rot, you will need to take the door off it's hinges, make a clean straight cut over the width of the door using a circular saw to remove the rotted area, and then piece in a strip of wood the size you need to match door width and whatever the dimension of the gap is, and glue and nail the strip onto the bottom edge of door...
There are door sweeps and weather stripping to seal gaps underneath. They also sell spray insulation that you can use, this product fills in any cracks and hardens. All of these products are $10 to $40, depending on the gap size.- www.homelyville.com/
First, before you attempt any fix, make sure you aren't dealing with termites or other pests that are weakening the wood. Also make sure you are not dealing with water damage and that there is mold. For sure then if it more than just a broken corners from wear and test or an accidental break, it will need replacement and/or if pests having the house checked and if needed, treated.
I have seen people fix broken corners of wood with Elmer's wood filler or Bondo or Bondo...can't remember exact name. There are YouTube videos that show various techniques. You need sandpaper and a putty knife, paint to match. Some techniques also use wood hardeners.
In a high traffic area this work is usually just a short term fix as often it will keep happening that the corner breaks off or there will be new breaks.
In the long run you are usually better off going to a place that sells salvage doors and getting one that fits and replacing it. Sometimes quick fixes are penny wise and pound foolish. Post back how it goes!
Go to a home improvement store and buy a door sweep that nails to the bottom of the door. We just did this to the door that goes into the garage and it made a huge difference