No.

The Mayan long count calendar is based on days, not solar cycles.. Because of that, leap days are irrelevant.

Mayan calendar basics

The long count calendar is based on a series of 5 numbers each representing a number of the previous period — usually 20, except for the tun period:

image from Wikipedia

2012

The 2012 Phenomenon is based on the idea that Mayans believed their current world (the fourth world) ended after 13 b'ak'tuns:

The Popol Vuh describes the gods first creating three failed worlds, followed by a successful fourth world in which humanity was placed. In the Maya Long Count, the previous world ended after 13 b'ak'tuns, or roughly 5,125 years.

0.0.0.0.0 is the beginning of the first b'ak'tun and 12.19.19.17.19 is the last day of the thirteenth b'ak'tun.

The full size of 13 b'ak'tuns — thirteen multiplied by 144,000 days — is 1,872,000. (This is 5128.8 periods of 365 days, so you can already see that the information cited by Wikipedia for a rough translation of solar years has taken leap days into account.)

To use this number of days to figure out the "end date", in other words, the day that corresponds to 12.19.19.17.19, all we need to know is what equivalent Gregorian calendar date corresponds to the Mayan date 0.0.0.0.0. And to know if December 21, 2012 is that date, all we need to do is figure out 1,872,000 days from the start date.

For that, we turn to this well-sourced article by John Major Jenkins, a student of Mayan time:

But how are we to relate this to a time frame we can understand? How does this Long Count relate to our Gregorian calendar? This problem of correlating Mayan time with "western" time has occupied Mayan scholars since the beginning. The standard question to answer became: what does 0.0.0.0.0 (the Long Count "beginning" point) equal in the Gregorian calendar? When this question is answered, archeological inscriptions can be put into their proper historical context and the end date of the 13-baktun cycle can be calculated. After years of considering data from varied fields such as astronomy, ethnography, archeology and iconography, J. Eric S. Thompson determined that 0.0.0.0.0 correponded to the Julian date 584283, which equals August 11th, 3114 B.C. in our Gregorian calendar. This means that the end date of 13.0.0.0.0, some 5125 years later, is December 21st, 2012 A.D.1

(emphasis added).

This same start date was also cited in Wikipedia

Assuming, of course, that the start date is correct, there is no way a lack of leap days in the Mayan calendar could confuse this at all. Leap days must be taken into consideration in the conversion, but historians have known to do that since the leap day was invented.

1,872,000 days since August 11, 3114 B.C. is a calculation that can be made safely with the calendar information we have. The best calculator I could find was this one, in which you can enter the dates and come up with a close (too close for leap days to affect it) but not exact result of 1,872,026.

You can also try this calculator which comes up with the same calculation using August 11, 3114 B.C., but if you enter the Julian date 584283, you get the exact period of 1,872,000 days until December 21, 2012.

Conclusion

So, there you have it — yes, the Mayan calendar did not include leap days. But, it didn't need to, since they didn't have anything to do with solar years. In fact, as you can see, the closest equivalent Mayan period to a solar year, a tun, is only 360 days — already five days off.