Emulsifying oil-soluble flavour compounds in homemade soft drinks

The emulsifiers are probably used to keep them mixed on the shelf, rather than to get them in to the mix in the first place. Traditional home made drinks such as still lemonade are intended to be mixed and served immediately, and plenty of stirring will disperse the oil in the water-based drink for long enough. Bottled drinks, whether ready to drink or concentrate, would separate in storage. There's also an aesthetic reason: people don't like to see a layer of oil on top of their drink.

Trying to extract the oils from the zest without steeping in further oils would be a good idea, because you'd have much less to emulsify or mix. Hot water would release quite a lot of the oil, and for some things an alcohol extraction would work (the amount of alcohol per serving would be miniscule if done properly).

In terms of replicating the commercial product you would be best off buying emulsifiers (lecithin perhaps). Reacting food grade sodium hydroxide with edible oils doesn't necessarily leave you with an edible product - saponification is a chemical reaction resulting in a new substance so we can't conclude it's food safe even if the reagents were. On top of this, soap in even tiny quantities is unpalatable.

There's so much rubbish written online about "natural" foods that it's hard to find much sense, but honey is reputed to have some emulsifying properties,and is a traditional ingredient in lemonade recipes.

To conclude, I'd try boiling the zest in water and straining, before mixing with the juice and water, then sweetening the lemonade with honey instead of sugar. Even if it doesn't work perfectly it should still be a pleasant experiment.