Can I use whole eggs instead of egg yolks in a lemon bundt cake?

In principle, you'd probably be able to find a recipe that uses ~2 eggs with a similar net volume, but it may not be so easy to retrofit your existing recipe, as it's reasonable to expect that the recipe is relying on some property of the yolk (emulsifying power, fat content, etc). You may find the result satisfying, but it is likely that you will produce a different texture than the recipe intended.

Given my experience with cake baking, I can say it's likely easier to substitute whole eggs for yolks than, for example, trying to use whole eggs in place of whites. Egg whites foamed into a meringue structure can be used to provide structure to a cake in place of leavening, for example, but whole eggs will not replicate that structure without being separated.

On the other hand, since an egg yolk does not generally provide structure as much as it does flavor, other than the extent which emulsifying power constitutes structure, it's fairly likely that you can get a pleasing cake that will perhaps be less dense than the egg yolk-only version.

Keep in mind that the egg white constitutes approximately 2/3 of the weight of an egg; the density should be similar, however. So it's quite possible you'll need only 4/3 eggs to provide similar volume of egg product; it's up to you whether you'd want to split the egg, round up to 2 eggs, or use one egg plus one yolk.

I should add, though, that there are plenty of ways to use up surplus egg whites if you make the recipe as written:

• Angel food cakes typically use egg whites and no yolks.
• Meringues, and variants such as marshmallows and French macarons (and American macaroons), use egg whites.
• Traditional sours, cocktails with lemon or lime, liquor, & bitters, typically use one egg white per serving; when shaken, the egg white foam adds a pleasant froth.

You can freeze egg whites in ice cube trays if, like me, your best intentions to use up the surplus are likely to be delayed for whatever reason.

The short answer is yes (although it would be less than 4 whole eggs as answered by @JasonTrue )

The change would be to the texture and flavour of the finished cake, and possibly the rise. Using yolks only will give a much denser, creamer, richer tasting cake. If you replace with whole eggs, you just get a regular cake. Personally, i'd keep the recipe as is and make macaroons - any excuse for coconut macaroons ;-)

One other difference - the yolks will add a very yellow colour to the cake - not sure how important presentation is to you, but its appearance will be altered if you swap out for whites.