How to make hydrogen peroxide into gel
There are a number of vegetable gums, such as carageenan, pectin and gum arabic, and man-made methyl cellulose which might be compatible with low concentrations of $\ce{H2O2}$, which you can determine easily. Any impurities, particularly transition metals, may make $\ce{H2O2}$ decompose, so it might be best to keep the gelling agent and peroxide separate until just before use.

If for oral use, this site suggests 0.75% to 1.5% solutions, one quarter and one half the strength of the common antiseptic N.B. $\ce{H2O2}$ bleach is much stronger and is not for oral use. Note that there are dangers with use of peroxide as a mouthwash.