Can I perform a stable isotope analysis at home?
I am assuming that you wish to measure isotopes of carbon.  It is unknown to me how you will extract sample from a bird feather that is suitable for measurement in a spectrometer, and does not taint the isotope ratios. The spectrometers that can distinguish between isotopes are generally quite sensitive and complicated. So the literal answer to your question is "NO, you can not do it at home with a DIY spectrometer."

But, you might be able to purchase a spectrometer. You might be able to differentiate carbon isotopes in carbonyl compounds with infrared spectroscopy. (But don't count on it, unless you are somehow making a gaseous product such as $\ce{CO}$ or $\ce{CO2}$.)

The usual way of measuring carbon isotopes is via mass spectrometry. You can probably get a crude measurement with a machine that costs $20,000 (on eBay) but then you must get it calibrated and keep it supplied with carrier gas, etc.  In short, it will bankrupt you.

User permeakra has the best idea: send a sample to a lab.  But, I am not sure of any lab will allow you to send samples through their spectrometer, not knowing anything about your reputation or the volume of business that they might expect to get from you.