613.7. Within a layer or sublayer, determining which order effects are applied in is sometimes done using a dependency system. If a dependency exists, it will override the timestamp system.
613.7a An effect is said to “depend on” another if (a) it’s applied in the same layer (and, if applicable, sublayer) as the other effect (see rules 613.1 and 613.3); (b) applying the other would change the text or the existence of the first effect, what it applies to, or what it does to any of the things it applies to; and (c) neither effect is from a characteristic-defining ability or both effects are from characteristic-defining abilities. Otherwise, the effect is considered to be independent of the other effect.
Since the Bear initially has no abilities, the "loses all abilities" part of Turn to Frog's effect can't do anything. However, once the Sword's effect grants the bear abilities, Turn to Frog's effect can remove those abilities. By that logic, it seems to me that Turn to Frog would depend on Sword of Vengeance.