Should I buy a blazer as an intern?

Agree that a blazer is generally useful, but "some business formal situations" means you will need at least one suit: if everyone else is in a suit, you will look under-dressed with blazer & tie.

For everyday wear, you will need a rotation of dress shirts fit your body, are made of quality fabrics. It turns out shirts different brands that have the same 15.5 X 34 label have wildly different sizing through the body, ranging from 48" of fabric around your belly button from Brooks Brothers Traditional to 37" for T.M. Lewin Super Fitted. If you'd rather not find your best fit via trial & error, there's https://www.wardrobeessentialist.com/post/how-to-find-your-next-dress-s…</a">an easier way to find a great fit for under $50 without ever setting foot in a store.

As an intern, your best approach will be to fit the style norms of the office. Solid blue and white spread collar shirts are "universal donors" which will work well in https://www.wardrobeessentialist.com/post/dress-for-the-destination-you…</a">a conservative boardroom, but also not look out of place in https://www.wardrobeessentialist.com/post/dress-for-the-destination-bus…</a">business casual settings. Start out with the basics, and then add patterns once you get a sense for the dynamic range of the group.