I'm a candidate for a data scientist position. Is it a good idea to scrape the company's website and run an analysis on the data to show my worth?

It really depends. Data scientists are relatively more expensive roles to fill relative to most other data engineers and analysts, and I would deploy them into specific points in data science projects with  measurable business uplift. As such, if I were you, I would spend my time understanding the profile and strategy of the business, thinking about which are good areas to delve into exploratory data analysis to initiate value creating applications of data science. If one of those applications is based on scraping data from the website, then great. If it isn't, then no. I would also create a small dummy data set, apply your data science skills to build a small mock implementation as a proof of concept, and tell the interviewer what a successful implementation would mean to the business. As an example, if the company is say, a retailer competing on the basis of customer service, I would build a little data frame or csv file of customers' logged feedback and attending service staff. Run some text analytics / categorisation to parse the feedback, and start picking out service staff who could require attention and calling out themes to address. Bonus if you can expound on how you would explore the links between customers service and churn. Double bonus points (and possibly a job offer on the spot if I were the employer) if you can sketch a business case on what are the tangible and intangible benefits of your work, how your algorithms will be embedded in the company and maintained on an ongoing basis, and how long it would take for it to pay for itself.

Sounds like a winning strategy. Good luck. P.S: http://automattic.com/work-with-us/data-wrangler/