This banana bread recipe calls for applesauce and it's not something that's available at the stores here. Will it be okay if I just leave it out or can I use any other common ingredient to replicate the effect?
Applesauce has little flavor and almost no nutritional value. It is added to baked goods for the one thing it does have- pectin.
Pectin interferes with gluten in a similar manner to fat although by a different mechanism. For this reason applesauce is often used to replace some of the fat in recipes for health-conscious bakers. See this question. Or if you are more ambitious.
Other banana bread recipes are very similar to cakes and rely on a huge amount of fat- often creamed with a large amount of sugar.
Your options are then to:
Why not making your own? Making your own applesauce is actually very easy. So I bet there are apples for sale in the stores? The simpelest I can think of is:
3/4 cup water
(1/4 cup sugar)
1/2 teaspoon ground cinnamon
Peel core and chop the apples. Cook it all together on medium heat until the apples are soft. (15-20 minutes)
It is no work at all and tastes delicious.
However if you think it is to much work anyways (but I do not hope so!) it is good to know that in lots of recipes, applesauce is used to replace oil to lower the amount of calories. (Some also claim that applesauce replaces eggs, but I have no experience with that) However, stating something like: 'replace 1 cup of applesauce by 1 cup of oil' can not be done properly. Applesauce contains water where oil does not. So it will make your cake more fat, and applesauce make the baking more 'cakier'. Also you are making a fruit cake, and of course the apple taste improves this fruity taste. I also think using oil will make you cake more 'heavy' then applesauce. You can read more about applesauce replacing oil via this link. I think if you are going to replace the applesauce by oil, you have to experiment about the substitution rate. But I would start with using 1/4 cup of oil in stand of the 1/4 cup of applesauce.
If you don't want to make your own applesauce, nor replacing the sauce by oil you might want to look at baby-food. Some jars of baby-food contain just some kind of apple puree.
The unsweetened applesauce also provides a bit of acid. How about replacing the same weight, not volume, with apple juice and increasing the banana content by an small amount -say half a sm banana?
According to http://www.onlineconversion.com/weight_volume_cooking.htm 1 cup AS = 8.6oz so 2 oz of juice would be about right
With only 1/4C in the recipe, I am assuming that changes here and there won't spell disaster.