The claim that a frog with eyes inside its mouth exists due to a rare genetic mutation is mostly true because multiple credible sources and documented cases confirm such anomalies, though the idea that it can see only when it opens its jaws is somewhat simplified and not fully verified.
Scientists discovered a frog with eyes inside its mouth, which is a rare genetic mutation that lets it see only when it opens its jaws.

Are These Frogs with Eyes Inside Their Mouths Real? - Snopes.com
Source: snopes.com
Key Evidence
Multiple sources including Snopes and scientific discussions confirm the existence of frogs with eyes inside their mouths due to macromutations.
A notable case from Ontario in 1992 is frequently cited as a rare but real example.
Functional claims about vision linked to jaw opening are mentioned but lack definitive scientific confirmation.
What the Evidence Shows
There are documented instances of frogs or toads exhibiting a rare genetic mutation known as a macromutation, which causes their eyes to develop inside their mouths rather than on top of their heads. This phenomenon has been reported in scientific literature and observed in nature, with at least one well-known case from Ontario in 1992. The mutation significantly alters the frog's anatomy, making it an extraordinary anomaly.
However, the functional claim that these frogs can see only when they open their mouths is less clear. Some sources note that frogs use their eyes to assist in swallowing by retracting them into their skulls, but this does not necessarily mean the eyes inside the mouth function normally for vision. Photographic evidence is limited and often black-and-white or anecdotal, and some reports caution that images circulating online may be misinterpreted or exaggerated.
Overall, the existence of such frogs is supported by multiple independent observations and scientific discussion, but the exact visual capability related to jaw movement remains uncertain.