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All steels have a modulus somewhere between 28 and 30 million psi. The stiffness of a frame with identical cross-sections will be effectively identical between stainless and mild steel. If the frame has different thicknesses or gusseting, etc... that will affect stiffness.

300 series stainless steels are considered soft (low yield strength), but so is the mild steel used in car frames. The yield strength of annealed 300 series stainless is usually somewhere around 35000 psi, while mild steel (A-36 or equivalent) is about 36000 psi. Note this is for annealed steels. If you cold work steels they will gain strength above the values presented here, but still the stiffness (modulus) does not increase.

Do not confuse strength with stiffness (modulus). All steels flex about the same amount for the same shape. High strength steels will flex much further before yielding (permanent deformation) or breaking. That is the only difference. Parts made from 4130 or 4340 (chrome-moly steels) will flex the same as those made from mild steel, up to the point where the mild steel part takes a permanent set. 4130 and 4340 are stronger, but not appreciably stiffer.

Identical design frames made from stainless and mild steel will have about the same stiffness (<5% variation).