Don't know specifically about Port of Tacoma, but I could not find anything in the Pierce County Code that would prohibit shooting a bow in the county for your specified conditions. However, you should satisfy yourself that that is the case, as well as whether there are any animal cruelty type of ordinances that might bear on what you want to do (this is not a moral judgment - just saying for your protection.
As a general proposition, counties and municipalities only prevent such archery activities within the parks under their authority. This is the case for my municipality (rather large one) and I would shoot a bow in my back yard after having search the code for such prohibitions.
For those of you wishing to check the city and county codes for similar questions, you can typically find them at the link below if your city and county participates. I don't know how stale this information gets. That is, I am not sure how frequently these are updated as the respective codes change.
http://www.mrsc.org/codes.aspxAbout MRSC
The Municipal Research and Services Center (MRSC) is a private, non-profit organization based in Seattle, Washington. Our mission is to promote excellence in Washington local government through professional consultation, research and information services. All of the information and research services described below are available free of charge to elected officials and staff of Washington city and county governments and public hospital districts. MRSC serves Washington local governments by providing: (1) Dependable advice from a multidisciplinary team of professional consultants; (2) A comprehensive Web site; (3) Access to thousands of sample documents; (4) Timely and informative print and electronic publications; and (5) Access to the largest local government library collection in the Northwest.
And of course there are the RCW and WAC.
http://apps.leg.wa.gov/rcw/http://apps.leg.wa.gov/wac/If your Google-fu is lacking, keep in mind that you can use Google to search the WAC and RCW (and likely the MRSC) by using the Google "site:" operator.
For instance, the search string "firearm site:http://apps.leg.wa.gov/rcw/" will return all instances at
http://apps.leg.wa.gov/rcw/ that have the word firearm.
As a further example, after following the MRSC for Tacoma I tried the search, "firearm site:http://cms.cityoftacoma.org/cityclerk/Files/MunicipalCode/", in Google. You could do the same for your terms of interest such as arrow, bow, archery, etc.
