Yes, BrowserStack will charge VAT for all customers in Switzerland. If you have provided a registered VAT number to BrowserStack, the same shall be printed on the invoice that would be issued to you, for claiming input tax credit with your local tax authorities.
Every customer is unique and so are their specific tax circumstances. The above stated information should not be considered as tax advice, but as a general overview of relevant tax rules. BrowserStack cannot provide you with tax advice for your individual situation, and therefore we strongly recommend speaking with a professional tax advisor for tailored advice.