Limescale is made of insoluble carbonates of calcium and magnesium; in other words, they are salts of carbonic acid. Salts undergo displacement reactions: for instance, weaker acids tend to be displaced from their salts by stronger acids. In case of limescale it means that it is decomposed by acids stronger than carbonic acid (acetic acid, the main constituent of vinegar, is technically weaker than carbonic acid - but carbonic acid is unstable and easily decomposes to carbon dioxide and water - taking that into account explains why carbonate anions are still being displaced by acetate anions). My point with this theoretical introduction is that even if there existed a type of soap that was safe for aquarium use, it wouldn't be the optimal solution for removing limescale and wouldn't do as good in removing limescale as vinegar. Soaps tend to be alkaline and alkaline is the opposite of acidic, so they cannot dissolve limescale the way vinegar does. Their mode of operation in cleaning is different, they act as surfactants and emulsifiers instead, which means that they facilitate dispersion of immiscible liquids in one another (like oil in water) - for this reason, they are much more suited for cleaning greasy stains based on fat.
Acids even stronger than vinegar - like muriatic acid from the hardware store - are theoretically even better in dissolving the limescale, but they are too dangerous to be used in aquarium: not only for the aquarium inhabitants, but also for people because they are corrosive to the skin and eyes; also, as far as I know they are of industrial-grade (as opposed to food-grade in case of vinegar), so they might potentially contain undesirable contaminants.
For these reasons, in my opinion there isn't a better limescale cleaning substance than vinegar.
Please also note that it is the safest to use white vinegar (which is made from distilled spirits). Other types of vinegar (apple cider vinegar, etc.) contain many additional compounds beyond acetic acid that could be potentially undesirable or harmful - also, we don't even need them for cleaning. Acetic acid isn't toxic per se, but it is best to limit the amount that spills into the aquarium water because it would have the side effect of altering water parameters (namely, it would lower pH and KH). It is also absolutely important to avoid having undiluted vinegar come in contact with the media inside the filter, because nitrifying bacteria inhabiting the media could be destroyed by its low pH; also, please avoid having undiluted vinegar come into contact with animals living in the aquarium, because it is corrosive and irritating to mucous membranes. If cleaning power has to be sacrificed for safety in sensitive places of aquarium, vinegar could always be diluted with water.
In terms of using safe tools for cleaning - you have touched an important subject there. It is indeed potentially unsafe to use normal kitchen sponges - even if the sponge is brand-new and doesn't have any dish soap residue because of having never been used before, the sponge's material itself might release chemicals potentially harmful to aquatic life. Sponges for kitchen use are made of two layers of different polymers united together - this results in a double-sided sponge with one softer side and one stiffer, abrasive side meant for scrubbing. The softer side is usually made of polyurethane foam; a nasty thing about polyurethane is that it releases extremely toxic fumes during combustion (of which the main concern is hydrogen cyanide and carbon monoxide) and, like most synthetic polymers, it is flammable - so for safety, it is often treated with flame retardants. However, almost all of the flame retardants themselves are considered harmful - and if they leach into the water, aquatic animals are much more vulnerable than humans.
I don't think it would necessarily be that immediately dangerous if a sponge like this was used to clean the aquarium; the much more hazardous scenario would involve using a sponge like this as a medium inside the aquarium filter. On the other hand, using it just occasionally to clean the aquarium would probably not result in any serious disaster. Nonetheless, it is still better to use safer alternative whenever possible.
I could suggest buying a sponge medium or filter floss for an aquarium filter and using it for scrubbing off the limescale. I am silently and naively assuming that polymers used for them are not treated with potentially harmful compounds, which might actually not be the case - but still, I consider them to be safer and I trust them much more than any materials not meant for aquarium use. Generally, materials and tools explicitly meant for being used in aquarium are safe; the tools from a hardware store, etc. might potentially cause unpleasant surprises. For example, sterile cotton balls might seem appropriate and safe for aquarium use, but as far as I know they may sometimes contain pesticide residues, especially the cheaper ones; that is because cotton is one of the most pesticide-dosed crops in the world. Such residues would be especially harmful for aquatic invertebrates, like shrimp.
In regard to which tools should be used, I could also suggest using a scrubber intended for removing algae from the glass. In my opinion, it would be the best to have a vinegar-soaked material somehow touching the limescale layer for an extended period of time, so that the vinegar's evaporation is slowed down and the limescale could "soak in" the vinegar - even if it wouldn't completely dissolve the limescale, it would definitely ease scrubbing it off later.