Neuropsychiatric assessment is a very large area. What are you looking for in particular? It's hard to see some areas being assessed successfully online too…
We're hoping to get some indication of anxiety and trauma-related disorders, depression (MDD, bipolar), Parkinson's disease, schizophrenia, autism spectrum. Based on that we plan to identify participants that will come in for formal clinical diagnoses.
You can use almost all questionnaires that you want (e.g., a validate questionnaire assessing neuropsychiatric disorder symptoms, like NPI (Neuropsychiatric Inventory) {depending on what you want to include} and implement them on web-surveys, such as surveymonkey (https://en.surveymonkey.com). You just need to copy-paste all questions from a questionnaire to this web survey. Then, you can send a link to many participants and collect all answers.