Can I print off my entire bank statements for the year and use all the purchases at stores off my statements too figure the sales taxes I've paid for the year?

Yes

if the bank statement shows what portion your purchase was applied to the sales tax,

or if you have some reasonable way of figuring that.

It seems like it might be pretty hard to do this since the bank statement does not show a break-down of the purchase. For instance, some states have different sales tax rates on food than merchandise.

"State, Local, and Foreign Income Taxes -- State and Local General Sales Taxes

State and local income taxes withheld from your wages during the year appear on your Form W-2 (PDF), Wage and Tax Statement. You can elect to deduct state and local general sales taxes instead of state and local income taxes, but you can't deduct both. If you elect to deduct state and local general sales taxes, you can use either your actual expenses or the optional sales tax tables. Refer to the Form 1040, Schedule A Instructions, for more information and for the optional sales tax tables. You may also use the Sales Tax Deduction Calculator."

https://www.irs.gov/taxtopics/tc503.html