The first thing you need to nail down is your research objective and how you will be analyzing your data then you conduct a power analysis.

See the attached paper and link. 

https://dgwinn.wordpress.com/2014/04/06/virtual-ecologist-approach-using-simulations-to-evaluating-experimental-designs/

It also depends on the scale of the project. Do you only care about the immediate area around each nest? Or are you more interested in larger scale habitat characters? One random plot per nest should be the minimum but it depends on the time and effort required to gather the data that is pertinent to the question. You should also think about what you will do if the random point lands you in an area that teals obviously do not use. How would you justify taking habitat selection measurements for a random point that placed you in the middle of a road?

The paper that Dan recommended is great. Another method is to sample as a percentage of your habitat of interest. The scale will indicate if that is 1% or 10% etc.

make sure you have a clear hypothesis to test: for eg. Wet sites have more aggregation of teals, the Null hypothesis, would be no diff in sites, then make sure u have a balanced design so equal no of wet and not wet sites randomly stratified by other covariates, eg. altitude, forest vegetation etc. Then conduct a power analysis to minimize your type-II error. Again, a lot will depend on your objective and whether you want a repeated measures test that may change year to year, so more sites vs more years is another question you may want to answer before you design your sampling program.

One last thought - it's probably not a good idea to automatically assume what's been done in the past is appropriate, but in some cases it can provide really nice guidance.  If I were you I'd read duck papers that address questions similar to yours and see if there is a sensible standard used by others that will indeed relate to the kind of analysis you're interested in doing.  With songbirds, for example, many people use a variant of the BBird protocol for measuring vegetation around nests and at random points.

In case of  open steppe small birds in Mongolia , especially for Alaudidae, our institute researcher or ornithologist identify and determine all species plants (geo-botanical records) around of the nest within 50 x 50 cm every nest they found. Geo-botanically, it should be 1 x 1 meter and I think  this  size would be better for teal.    

Try to start with small areas and characterize the vegetation. If you continue to find new plant species enlarged the area until you dont get new floral species. Then you can start your bird survey.

I think everything that has been mentioned is great advice Casey. The paper Dan posted is a great starting point, and as Andrew mentioned, using previous studies can often be insightful. Be sure you continually look at your data as you are gathering it through the field season. Look at the variation in your data and adjust accordingly for the next season if needed. A lot will be determined by what metrics you plan on taking at each veg point.

Finally, be mindful of what you are calling habitat. Remember that by definition habitat includes food, water, and cover and is organism specific. So keep that in mind when you are mentioning sampling in different "habitat", and make sure that terminology makes sense. Just a word of advice for a future defense ;) Here is a paper that helps bring this point to light.