TL;DR; There are studies that do show at least some correlation between lunar cycles and germination and plant life cycles. However, the correlation is rather complex, and apparently species-dependent.
A quick google search for this finds many references to university studies on the topic, allegedly done by Wichita State University and Tulane University. The oft-quoted paragraph reads:
Other experiments have been conducted at Wichita State University and at Tulane University. All have achieved the same results. Experimentation indicates that seeds sown just before or around the full moon have a higher rate and speed of germination than those sown at the new moon because seeds are able to absorb more water at the full moon.
I have yet to find any specific reference to the Wichita State study, however the Tulane study is easier to come by, although I have yet to find an online copy. (source)
A few other referenced (but unconfirmed) claims include:
Metabolism, water absorption, fertility and germination all are reported to respond to this synodic cycle (Graviou, E.. 'Analogies between Rhythm, in Plant Material in Atmospheric Pressure and Solar-Lunar Periodicities'. International Journal of Biometeorology, 1978. Vol. 22, No.2.). North-western University in Illinois demonstrated 35% higher water absorption in beans just before full Moon compared to new Moon. (Brown, F., & Chow, C., 'Lunar-correlated Variations in water uptake by Bean seeds', Biological Bulletin, Oct., 1973, 145, 265-278. ) Confirmation of this was attested by Dr Jane Panzer of Tulane University. (Panzer. J. J., 'Lunar Correlated Variations in Water Uptake and Germination in 3 Species of Seeds', PhD. U. of Tulane. 1976) Interestingly these remained in a diminished form if the pinto beans were sterilised, and further diminished by pasteurisation. Germination also showed this monthly rhythm in her studies (source)
I have thus far found only one original study on this topic, entitled Chronobiology of trees: Synthesis of traditional phytopractices and scientific research as tools of future forestry, and it details experiments with germination of seeds at various times of the lunar calendar. The study was conducted in 1990 and 1991 (with a brief preliminary study in 1989). A few relevant excerpts (emphasis added):
The speed of germination or beginning of emergence already shows a significant
difference between the [full moon] and [new moon] sowings for all the sowings and especially for those corresponding to the dry season, which took place in the middle of the trial. In mean values for the whole trial, the FM seeds germinated after 47.5 days, 19 per cent faster than the NM seeds, which appeared after 58.5 days. Experiments on the radish (Raphanus sativus) by Fritz (1994) show the same tendency: faster germination for
sowings shortly before FM.
These studies seem to agree with the hypothesis that the cytokinine content of
plants is linked to the synodic lunar rhythm, with a maximum at full moon (this had
been shown by Hofman, Featonby-Smith and Van Staden on algae in 1986 – cited in
Fritz, 1994). Cytokinine also plays a role in the model proposed by Rossignol et al
(1990) to explain the variations in the relative frequency of three forms of DNA
according to lunar phases.
These differences in the speed of germination are probably also partly linked to
cyclic variations in the absorption of water by seeds, as shown by Brown and Chow
(1973) working on a large scale: 7,931 series of 20 beans. One of the absorption
maxima coincides with the FM or shortly before (see Figure 1).
Water absorption by bean seeds (Phaseolus vulgaris) and lunar phases.
Note: A, B and C are the three different periods in 1972 and 1973. 3-day mobile means.
Source: Brown and Chow, 1973
And from the report's Conclusion and outlook:
These trials make clear, for the first time in trees or shrubs, the existence of a real
phenomenon, often mentioned in traditions or issuing from empirical experience,
consisting of a link between the lunar phases (synodic rhythm) and behaviour at
germination and during initial growth. They demonstrate that the phenomenon is not
as simple as it might seem at the outset, going beyond the general ‘cause and effect’
model and calling on predispositions or types of reaction specific to plants themselves.
These trials in turn raise questions about the exact nature of this phenomenon and of
the physiological processes involved.