Bridge rectifiers are so cheap these days that everyone should have a bunch in their inventory.
ex.
https://www.digikey.com/product-detail/en/smc-diode-solutions/UG3KB60GTB/1655-1838-ND/7244866Added bonus... some have hole in the package allowing for easy heatsinking :
https://www.digikey.com/product-detail/en/micro-commercial-co/GBU4J-BP/GBU4J-BPMS-ND/1140592So 2nd bonus, being glass passivated, they can be heatsinked on same heatsink with the regulator ... which could be the metal case itself.
And while you're there, you could easily replace the fixed voltage regulator with an adjustable one and either use a switch to fix between some preset voltages, or use a potentiometer.
ex LM317, 1117 regulators, 1085 or 1084 (lm1085, ams1084, whatever) etc etc
and depending on input voltage you may not need such a big capacitor.
ex for 9v output with 7805, you need a minimum of around 10.5v at input to get 9v at 1A at output .
If you have a 12v ac transformer , after rectification you have a Vdc peak of 1.414 x Vac - 2x voltage drop on diodes, so 12v x 1.414 - 1.5v = ... let's say 15v
So you can approximate capacitance required with formula C = Current / [ 2 x AC freq x ( Vdc peak - Vdc min) ]
For 1A and 10.5v min voltage, you have C = 1A / 120 x (15-10.5) = 1/540 = 0.00185185 Farads or 1850 uF ... so basically for 1A, a 2200uF 25v capacitor would work in this scenario.
Doesn't hurt to add a power on led, if only to be a minimum load and help the regulator give a better output.
Also wouldn't hurt to add some decoupling capacitors .. add a ceramic 0.1 uF capacitor as close to the input voltage pin as possible. Maybe add a 1uF ceramic on the output, besides the regular electrolytic capacitor.