Speech emotion recognition (SER) is an important research topic in human-computer interaction. Existing works mainly rely on human expertise to design models. Despite their success, different datasets often require distinct structures and hyperparameters. Searching for an optimal model for each dataset is time-consuming and labor-intensive. To address this problem, we propose a two-stream neural architecture search (NAS) based framework, called \enquote{EmotionNAS}. Specifically, we take two-stream features (i.e., handcrafted and deep features) as the inputs, followed by NAS to search for the optimal structure for each stream. Furthermore, we incorporate complementary information in different streams through an efficient information supplement module. Experimental results demonstrate that our method outperforms existing manually-designed and NAS-based models, setting the new state-of-the-art record.