Is there evidence for time varying second moments in annual economic data?
Yes, although not that much in finance in particular but in economics in general resounding yes. For example, the highly cited Engle (2001), GARCH 101: The use of ARCH/GARCH models in applied econometrics, besides examples with daily data refers also to some examples with quarterly data. Or more directly an example of this that actually uses low frequency data (yearly/quarterly/monthly) and uses variant of GARCH, GARCH-SK, is Narayan and Liu (2018). Or another direct example is Kontonikas (2004) who applies GARCH-M models to quarterly and monthly inflation data. So GARCH is and can be to many economic series on quarterly or annual frequency.
Usually people go for quarterly data or yearly data when it comes to macro variables like GDP, unemployment, inflation etc and for higher frequencies when it comes to finance (e.g. stock prices, bitcoin etc). This is because typically having high frequency means more data. However, at the same time having more data is not always better. For example, having access to GDP data on daily frequency would likely just add noise so there is no point in trying to go for high frequency data there. However, the same does not necessarily hold for variables like exchange rates or stock prices. What the optimal frequency for particular research is has to be to an extent decided on case by case basis. Also GARCH models are somewhat more common with high frequency data as they are primarily used to solve issues like volatility clustering, which will be more common problem at higher frequencies, but as the references above show application to lower frequency data is not unheard of.
What lags are commonly used in this case?
Lags in GARCH models do not depend on frequency of data per se (of course it is more likely there will be more autocorrelation in high frequency data than in low frequency data). You include lags in order to appropriately model autocorrelation. This will be case dependent whether you have high frequency data or low frequency data. This being said evidence shows that often it is hard to beat GARCH(1,1) (see Hansen et al. 2001) and this often holds for various frequencies*, but if you would want to do it 'by the books' you should just test for this and in addition you can potentially estimate several models with different $p$ (and $q$ as well even if that was not part of your question) and compare their performance (it is really not that much extra work once you write code for the main model).
* Although note the paper has received some criticism and so you should not just necessarily take it at face value - see here