Hello Jamie,
First, as you probably know, many mushrooms are poisonous, so you should never eat a mushroom found in the landscape or forest unless you are confident that you can positively identify the type. Most mushrooms do not harm plants or the soil, so it's okay to plant vegetables in that area. 

The following is an excerpt from Iowa State University Extension:

Mushrooms are the reproductive or fruiting structures of fungi. Their appearance in the lawn is usually indicative of decaying tree stumps or roots in the soil. In garden beds, mushrooms can appear because they are associated with decaying organic matter which could be dead roots, stumps, or mulch. Mushrooms typically appear when the environmental conditions are ideal for their development. For many mushrooms, this is cool and damp.
https://yardandgarden.extension.iastate.edu/faq/how-do-i-get-rid-mushrooms-my-yard-or-garden-bed

You can leave the mushrooms alone and they will disappear when the weather is warm and dry. However, the spores remain in the soil until the rotted material they live on is gone. They will reappear when weather conditions are suitable. Also if you wish, you can mow the area or rake the mushrooms out and dispose.

This article from Texas A&M Extension also gives good information about how to deal with mushrooms
https://plantclinic.tamu.edu/2011/04/25/fungi-in-the-garden-beds/

A final thought: What weed control is best in the garden? The following articles give great advice about what to use for weed control in the garden, both organic and non-organic. https://www.canr.msu.edu/news/smart_gardeners_improve_their_soil_and_weed_control_with_organic_mulch 
https://www.pubs.ext.vt.edu/426/426-326/426-326.html

I hope these articles help answer your questions, but please contact us again if you have further questions or need more information.

Happy Spring!