Can an iodine clock reaction work without using a strong acid?
You wrote the answer yourself:  the hydrogen peroxide consumes hydrogen ion according to the reaction

$\ce{H2O2 + 2 I^- + 2\color{blue}{H^+} -&gt; I2 + 2 H2O}$

You need the acid to provide the hydrogen ions, especially since if you allow the solution to become basic then the iodine disproportionates (see "Chemistry and Compounds" section, "Iodine Oxides and Oxoacids" subsection).

In terms of pure chemistry, maybe you could have used a moderately strong acid such as phosphoric acid or a bisulfate salt.  But then the limited dissociation of such species might make the iodine formation reaction slower and thus slow down the "clock".

Looking at the Wikipedia article on this reaction, one alternative is to use a peroxydisulfate salt instead of hydrogen peroxide as your oxidant.  This reacts with iodide ion according to

$\ce{S2O8^{2-} + 2I^- -&gt; 2 SO4^{2-} +I2}$

with thiosulfate again as the reducing agent to regenerate iodide ion.  Unlike hydrogen peroxide or the other oxidants, the above reaction does not consume hydrogen ions and is your best bet to run the reaction without a strong acid addition.  However, peroxydisulfates have their own hazards (one of which is requiring a strong sulfuric acid solution to make it in the first place), and if you make the common choice of potassium peroxydisulfate rather than the ammonium salt you could have solubility issues.