The degree to which people respond to a renewal form checkoff depends first and foremost on the level of awareness and positioning of your Annual Fund. If it's well understood what the Annual Fund does and it's seen as legitimate and unique, then a check-box by itself can be very successful: a service "that needs no introduction" is the only kind that tends to perform well on an automatic marketing vehicle (such as a renewal form that allows no space to provide a proper introduction). If it's not well-understood, consider testing a buckslip with one renewal effort that helps to explain what the Annual Fund has done, and what it could do with more support. Keep it to just one effort for starters and allow some time before evaluating the results.
If many of your renewing members go online to close the deal, you might include an alternative checkoff there if it can be programmed in what remains a very clean, neat Member Application. But if you have to direct them to a second donor screen to complete the transaction, I wouldn't—it's again a distraction from the membership renewal. As an alternative, choose a logical time to mount a dedicated fundraising effort among renewing members, say 4 months into the new membership year for that individual. A dedicated effort will effectively cultivate and fundraise to the prospective donor.
Our fundraising work has shown quite a few charities who generate 15+% of their total direct response funds through "second asks" (i.e. gifts made in response to a solicitation included with the donor acknowledgement/confirmation form for their first gift). So we know that good donors will give frequently, but asking for a gift within seconds of the online member renewal will get a response that generates a small fraction of what you can generate with subsequent targeted efforts. It makes sense to allow sufficient space and time between efforts to cultivate, perhaps suggest and explain other successful fundraising techniques such as matching gifts, restricted fund solicitation, major/planned giving inquiries, and continuity giving through EFT or credit card.
By far the best performance achieved by associations is with forms that are prepopulated with a suggested gift that the renewing member must consciously omit if they don't want to give. Ideally a personalized gift array featuring their highest previous contribution + a small multiple to upgrade them, or a flat amount slightly higher than the historical average gift made by all donors, if you can't do that. If the vast majority of your members are relatively indifferent toward your Annual Fund as a cause, a large proportion of them who currently don’t take the active step of opting into the forms you used in the past also won't take the active step of opting out of making a reasonable donation. You could easily go from 4% of members donating with a $25 average gift to a 40% rate with a $35 average gift. This would be an impressive increase in the proportion who give, but of course it's still not much money: $50,000 with 5000 members for example.
Bottom line, effective association foundation fundraising often begins with managing your checkoffs (not your Checkovs!), consciously and effectively.