Is there a standard way to write this when limited to ASCII?
For example, Danish/Norwegian replace æ, ø, å with ae, oe, aa. It seems less likely that something could exist and be reasonably readable for Polish.
Most search engines find "Zazolc" and "Zażółć" equal because of that. This becomes a problem in case of the words like "paczki" (boxes) and "pączki" (donuts), which have their own separate meaning - as explained in the article.
In contrast to most European countries, in Poland we use American keyboard layout with "Polish (programmers) layout" keyboard setting in OS.
You press ALT+A, ALT+E, ALT+L, ALT+S, ALT+C, ALT+Z, ALT+X to write "ą", "ę", "ł", "ś", "ć", "ż", "ź", respectively.
Some words written without Polish characters can become ambiguous without context. For example: word "łaska" - "mercy", written without Polish letter "ł" is "laska" - "stick".
Although, in old good times some people used British pound character in texts to express this letter, since '£' is visually similar to 'Ł' and more often available.
For other characters (ś,ź,ć,ż,ó,ą,ę) nothing like that was widely adopted. Workaround would be possible for 'ż' and 'ó' since they are (almost - see below) phonetically identical with 'rz' and 'u' respectively, but it wasn't popular, since most probably would be perceived as sign of very, very bad orthography.
*Almost, since some people claim they can distinguish these, but it's not popular ability.
When used in (computer) writing this is very readable, but no one would do this with handwriting, I think most common cases nowadays would be SMS messages (especially on dump phones) or some weird displays that are unable to properly render diacritic letters.