Can you keep your fresh cut herbs in plastic containers? Yes and No!
You can keep them in anything that you can afford. Glass jars allow things to remain sealed but not get crushed or damaged. They are also nonporous and don't absorb any of the last thing that was in it so it makes them reusable. With plastic containers you can store them, as well, as long as they are already dried completely. Plastic seals very well and if the herbs are still not fully dried you run the risk of bacteria growing in the sealed environment. They can work just as good as glass containers in that they don't allow moisture back in and that they provide a crush proof container. However some plastics have been shown to leak a toxic chemical into things that are stored within them, which is why I say glass is better.
Paper or cardboard allows herbs a place to dry because the paper fibers absorb moisture from them. However, they shouldn't be reused for a different herb because the fibers also absorb a lot of the one herb that was used for them. When I dry my herbs and I can't lay them out flat in the open to dry I place their leaves into paper envelopes and file them inside a cardboard box, or I just place the herb whole inside the box. This allows for better storage and space conservation. Paper products should not be used for long term storage of herbs because the drying process never halts and continues to degrade the product. Sealed glass containers (or plastic) allow the dried herb to be stored while maintaining a level of freshness longer.
When making essential oils or any thing with an oily base I never use plastic. Not only is there the risk of the chemical contamination but also oils tend to degrade the plastic and eventually will leak. Dark colored glass is best to keep light from damaging the herbs or oils as well. Dark amber colored or dark blue work the best because they diffuse the damaging light rays.
You can find many great affordable glass storage containers at your local craft store or grocery store. Whether it be clear glass Mason jars or decorative fancy bottles the most important thing is to protect them from light. If you can, try to find a chest or cabinet to place them in that closes out the light. Never store them in direct sun light once they are dried. Using the sun to expedite the drying is fine but once they are dry they must be placed out of the light to protect their freshness.