Vegan egg substitute in sweet roll dough

You can use either chia seeds or ground linseed. Mix a tablespoon of chia/linseed with 3 tablespoons of water and let it sit on the counter for about twenty minutes.

That mixture will replace the egg in almost any recipe for bread/mixed cake.

The rest period is important, as it allows the water to soak up the sticky proteins from the linseed/chia.

Going out on a limb here, because I'm not a bread expert, but since your recipe includes plenty of water, yeast and flour, which are known --by themselves-- to produce delicious results, my feeling is 1 egg can safely be left out without causing a massive disruption in the balance of the rest of the ingredients. Common sense tells me any loss of richness from one missing egg will be barely noticed in a mixture with 6.5 to 7 cups of flour.

Eggs add oily richness, so sometimes it helps to throw in a teaspoon or so of extra vegetable oil to help with that homey yummy "oomph", but in this case, the caring gesture of inclusiveness toward the new members of your family will probably more than compensate for the loss of an egg. (My 2 cents worth.)

Soy flour, chickpea flour or other legume flours (mixed in with the main flour at 1:20 to 1:5 by weight) work well for some recipes - experimentation needed.

These are strong,sticky binders (a slurry of chickpea flour can be turned into a viable omelette); how they will effect the final texture is quite dependent on leavening methods etc.