My understanding is that no plant contains the full spectrum of amino acids. Would it be possible to take certain amino acids while eating plant foods to “fill in the gaps” and make the entire plant food usefull as protein?
You can get the “full spectrum” of essential amino acids by very carefully choosing a varied diet. While most plants don’t have all of them, there are combinations of plants that fit the bill (No examples, but Google should turn something up). THere are a few plants with complete amino acids, including (IIRC) peanuts, soybeans, lentils, and beans. There are likely others. Consult a dietitian or other medical professional before embarking on any radical dietary change.
To my knowledge, no legume except soy provides the complete spectrum of necessary amino acids. However, any bean plus any grain provides complete protein. Still, my understanding is that the ratios are not as ideal as those provided by eating meat or animal products. Thus, a perfectly vegan diet requires considerable effort to obtain decent protein. In less extreme diets, there’s very little risk, as most Americans consume far more protein than necessary.
You might want to pick up a copy of The Macrobiotic Diet.
Depending on who you ask, there are eight or nine essential amino acids, which are the amino acids that the human body cannot produce.
Soybeans (edamame, tofu, soy nuts) and quinoa (cooks like a grain, but it’s actually the seed of a plant that’s related to spinach) are complete proteins, meaning that each one provides all the essential amino acids on its own.
However, most plant foods are incomplete proteins because they are low in one or more essential amino acids. In particular, grains are rich in all the essential amino acids except lysine. However, legumes are rich in lysine. Legumes are low in tryptophan, but grains are rich in tryptophan. That means that you can combine a grain (rice, wheat, corn, oats, etc.) with a legume (beans, lentils, peanuts, etc.) you’ve got a complete protein.
Examples would be a peanut butter sandwich, red beans and rice, bulghur and lentils, etc.
You might be able to do the same thing with supplements, there are much more delicious ways to do it. 
[nitpick]
All plants contain all 20 common amino acids.
Few plants contain them in the proportions needed to fuel active omnivores.
[/nitpick]
[nitpick nitpick]Many if not most active omnivores survive quite well on single plant diets because they utilise gut bacteria to manufacture the lacking amino acids. Humans have such ludicrously short guts and atrophied ceaca that we can’t manage that trick and have to get most of our essential nutrient direct from the soucre.[/nitpick nitpick]
Further information on amino acids:
I’ll assume it is common knowledge that amino acids are required for new cell growth. However, what is generally less well known is that in order to assemble a protein from amino acids, all of the requisite amino acids must be available and in sufficient quantities for the process to occur. If any of the 9 essential amino acids is missing or is not present in sufficient quantity, protein synthesis will not happen. This is called malnutrition and it is the source of much illness and death in the world today.
Be very careful about embarking on a vegan diet without a firm understanding of nutrition. Or don’t. Everyone is going to die anyway, right?
What happens to the incomplete amino acids? are they excreted as waste?
They are not incomplete, there are simply not enough of them. And yes, (IIRC)your body does not have the ability to store significant quantities of these substances and they are secreted as waste.
Sorry, the i know longer have the reference text I obtained this from (Nutrition and Mental Health, University of Idaho Press), and my memory may be less than perfect on some of the fine details. However, the upshot is that 4 ounces of fish once a day will pretty much fulfill all your body’s requirements for protein.
I should preview more often. Substitute excreted for secreted in the first paragraph and I no longer for that train wreck at the beginning of the second paragraph.
The body doesn’t waste amino acids by excreting them. If they aren’t used for protein synthesis for whatever reason the body simply strips off the amine group and burns them for energy. That’s why even the leanest meat is a perfcetly acceptable energy source. The amines themselves are converted to urea and excreted in the urine.