How is it possible to dissolve zinc oxide safely for skin use, preferably without acid?
As I previously stated in a comment, I think that the zinc oxide is not actually dissolved. That would just leave ions which you would hardly consider zinc oxide any more. $$\ce{ZnO + 2HCl &lt;=&gt; Zn^2+ (aq) + 2Cl- (aq) + H2O}$$

Instead I think that there are nanoparticles in use, that appear to be dissolved. In deodorant or other skin products it is likely to have quite a low concentration. 

An admittedly short search turned up a patent by Marc Paye, in which is stated:[1]


  [...]
  (f) 0.01% to 1%, more preferably 0.03% to 0.5% of Nanoparticulates of zinc oxide which functions as an anti-irritant system for anionic surfactants,
  [...] 


Or from another patent of the same company:[2]


  [...]
  (c) 0.5-10% (particularly 0.5-7.0, and more particularly 0.5-5.0%) of a small particle size zinc oxide, particularly a micronized zinc oxide or a nanoparticle size zinc oxide having a particle size in the range of 20 nanometers-200 microns;
  [...]


I also found, but have not completely read an article that confirmed my theory.[3]


  As an ingredient in dry deodorants to reduce wetness under the arm, ZnO can be used between 0.05 and 10% by weight with average particle size in the range of 0.02–200 microns. ZnO may be used to provide a pH range desirable for deodorants designed for use on sensitive skins [174].


At least in the last article you should also find some information about how to synthesise these particles. Spoiler: It's nothing you would try at home.




Paye, M.; Zinc oxide containing surfactant solution. US6774096 B1, 2004.
Hall-Puzio, P.A.; Gale, A.E.V.; Brahms, J.C.F.; Deodorant with small particle zinc oxide. US 6358499 B2, 2002.
Moezzi, A.; McDonagh, A. M.; Cortie, M.B.; Chem. Eng. J., 2012, 185–186, 1-22..