Emerging transportation technologies have the potential to significantly reshape the transportation systems and household vehicle ownership. Key among these transportation technologies are the autonomous vehicles, particularly when introduced in shared vehicle fleets. In this paper, we focus on the potential impact that fleets of shared autonomous vehicles might have on household vehicle ownership. To obtain initial insights into this issue, we asked a sample of university personnel and members of the American Automobile Association as to how likely they would consider relinquishing one of their household's personal vehicles if shared autonomous vehicles were available (thus reducing their household vehicle ownership level by one). For single-vehicle households, this would be relinquishing their only vehicle, and for multivehicle households (households owning two or more vehicles) this would be relinquishing just one of their vehicles. Possible responses to the question about relinquishing a household vehicle if shared autonomous vehicles are present are: extremely unlikely, unlikely, unsure, likely, and extremely likely. To determine the factors that influence this response, random parameters ordered probit models are estimated to account for the likelihood that considerable unobserved heterogeneity is likely to be present in the data. The findings show that a wide range of socioeconomic factors affects people's likelihood of vehicle relinquishment in the presence of shared autonomous vehicles. Key among these are gender effects, generational elements, commuting patterns, and respondents' vehicle crash history and experiences. While people's opinions of shared autonomous vehicles are evolving with the continual introduction of new autonomous vehicle technologies and shifting travel behavior, the results of this study provide important initial insights into the likely effects of shared autonomous vehicles on household vehicle ownership.