Food texture plays an important role in food acceptance by young children, especially during the complementary feeding period. The factors driving infant acceptance of a variety of food textures are not well-known. This study summarizes maternal reports of children’s ability to eat foods of different textures (here: acceptance) and associated factors. Mothers of 4- to 36-month-old children (n = 2,999) answered an online survey listing 188 food-texture combinations representing three texture levels: purees (T1), soft small pieces (T2), hard/large pieces, and double textures (T3). For each offered combination, they reported whether it was spat out or eaten with or without difficulty by the child. A global food texture acceptance score (TextAcc) was calculated for each child as an indicator of their ability to eat the offered textured foods. The results were computed by age class from 4–5 to 30–36 months. The ability to eat foods without difficulty increased with age and was ranked as follows: T1> T2 > T3 at all ages. TextAcc was positively associated with exposure to T2 (in the age classes between 6 and 18 months old) and T3 (6–29 months) and negatively associated with exposure to T1 (9–36 months). Children’s developmental characteristics, as well as maternal feeding practices and feelings with regard to the introduction of solids, were associated with texture acceptance either directly or indirectly by modulating exposure. Children’s ability to eat with their fingers, gagging frequency, and to a lesser extent, dentition as well as maternal feelings with regard to the introduction of solids were the major factors associated with acceptance. This survey provides a detailed description of the development of food texture acceptance over the complementary feeding period, confirms the importance of exposure to a variety of textures and identifies a number of additional person-related associated factors.