訴{うった}える does not derive from noun 訴{うった}え, but rather the opposite -- the noun 訴{うった}え derives from verb 訴{うった}える, as the stem or continuative form of the verb.
In turn, 訴{うった}える comes from older classical form 訴{うった}ふ. This was a 下{しも}二段{にだん}活用{かつよう}動詞{どうし}, or "lower bigrade conjugation verb". The "lower" part refers to the vowel of the stem, and "bigrade" means that there were two different vowel endings. For 下{しも} verbs, these vowel endings were ‑u and ‑e. (JA Wikipedia article here.) This contrasted with 上{かみ}二段{にだん}活用{かつよう}動詞{どうし}, for which the two vowel endings were ‑u and ‑i. (JA Wikipedia article here.) Over time (some time around the Muromachi period, 1336–1573) both bigrade forms collapsed to unigrade forms, losing the ‑u verb-stem endings to just have ‑e for 下{しも} verbs like 訴{うった}ふ (now 訴{うった}える) and 食{た}ぶ (now 食{た}べる), and to just have ‑i for 上{かみ} verbs like 落{お}つ (now 落{お}ちる) and 過{す}ぐ (now 過{す}ぎる).
Looking back further, 訴{うった}ふ got its geminated ‑tt‑ as a phonological contraction or erosion of older form 訴{うるた}ふ, which was apparently the form in use in the early Heian period (794–1185).
I cannot find much beyond this point in history to divine the verb's derivation any further. This verb urutapu is long enough to suggest a compound derivation, but I cannot find any clear roots. The verb ending in classical Japanese and earlier was ふ, a common auxiliary verb used to indicate repeated action or ongoing state, but this only attached to the incomplete form (未然形{みぜんけい}) of the preceding verb, and I cannot find any clear indication of an ancient verb urutu. The last two morae, たふ, could have been ancient verb 給{た}ふ used as an honorific auxiliary, but this only attached to the continuative form of a verb, and うる does not conform to continuative conjugation patterns. The underlying ultimate derivation appears to be lost in time.
(Note: Shogakukan's big 国語{こくご}大{だい}辞典{じてん} is my primary source.)