Is the first image of Psalm 29:9 one pertaining to flora (e.g. NET's "trees"), or fauna (e.g., ESV's "deer")? Good question!
The Septuagint took the key word as "deer" (ἔλαφος) here, and the KJV, and those deriving from it (essentially the "Revised" version descendents) have taken it as a "fauna" image, too, thus KJV:
The voice of the LORD maketh the hinds to calve, and discovereth the forests: and in his temple doth every one speak of his glory.
It is, then, just the first part of the verse we're concerned with; 29:9a runs in Hebrew:
קוֹל יְהוָה יְחוֹלֵל אַיָּלוֹת
qôl YHWH yəḥôlēl ʾayyālôt
There is a related thought in the LORD's speech to Job, in Job 39:1, when the LORD asks Job:
Do you know when the mountain goats give birth?
Do you observe the calving of the deer? [NASB]
In the context of Ps 29:9, however, the pairing of deer giving birth with stripping forests bare has often been thought to make poor sense -- if any sense at all. That both images are at home in the poem as a whole is clear enough: we get flora in v. 5, and fauna in v. 6, after all. But there the poetic pairs are consistent. And so, think some, they should here as well.
The second note on this verse in the NET Bible explains their justification for making the imagery consistent in this verse, levelling it to "flora", and doing away with the "fauna":
- "large trees" | v9/n2 tc Heb “the deer.” Preserving this reading, some translate the preceding verb, “causes [the deer] to give premature birth” (cf. NEB, NASB). But the Polel of חוּל/חִיל (ḥûl/ḥîl) means “give birth,” not “cause to give birth,” and the statement “the Lord’s shout gives birth to deer” is absurd. In light of the parallelism (note “forests” in the next line) and v. 5, it is preferable to emend אַיָּלוֹת (ʾayyālôt, “deer”) to אֵילוֹת (ʾêlôt, “large trees”) understanding the latter as an alternate form of the usual plural form אַיָּלִים (ʾayyālîm).
In other words,
- they find a problem with the semantics of the verb, taking the Hebrew at face value;
- they propose (as others have -- starting, I believe, with Lowth in the 18th Century) a convenient emendation of the word usually translated "deer" allowing a translation of "forests";
- and this makes the verse (they argue) sensible, coherent, and lexically sound.
However, there are some fairly strong arguments against this view:
- It isn't clear how the emendation to "forests" works with the verb ḥûl/ḥîl which means, as NET notes, to "give birth" (see bottom p. 297a for relevant BDB entry). The NET notes don't justify their semantic choice of "bend" here. The typical range of meaning given for this verb in the lexica has to do with "labour [of birth pains], writhing, trembling".
- Contrary to the NET note, some suggest (cf. BDB) that the use of this verb in this form (po'lēl) is, in fact, causative here (see "postscript" at the end of this answer for more on this).
- The implication of Job 39:1-3, noted above, is that the LORD does indeed bring forth the birthing of wild animals -- although the sense there is clearly "providential". One commentator (Baethgen, 19th C. German) notes, however, 1 Sam 4:19 where the shock of news brings Eli's daughter-in-law to birth, suggesting an analogy to the event envisaged in Ps 29:9a.
- As many note, no ancient version (Greek, Aramaic, Syriac, Latin) differs substantially from the Hebrew sense here, at least as regards flora vs. fauna.
- Of the commentators I've checked (Baethgen, Anderson, Dahood, Craigie, Kraus, Terrien), not one reads with the emendation ("forests"); each retains the Masoretic Text's plain sense ("deer" or the like).
Summary -- the latter considerations incline judgment toward retaining the sense conveyed in the ESV, as cited by OP, and rejecting the emendment offered by the NET translation.
Postscript on the so-called poʿlēl
As noted in the answer above the key verb in this question is usually taken to be form of the "hollow" root (middle-waw or -yod), ḥûl/ḥîl, sometimes noted as ע״ו. Verbs of this class (generally) don't show the characteristic doubling of the middle radical seen in the Piel and related (Pual and Hitpael), but they do show the doubling of their final radical in a formation not seen in other verb classes. This being the case, many grammarians believe that the poʿlēl and its related binyanim are the equivalent of the Piel forms for these "hollow" roots. See, e.g. GKC §55c and §72m; also the "Binyanim: Biblical Hebrew" article of the EHLL.
Although the origins and semantics of these forms are still debated, they are usually thought to be related to the Piel, and so are understood to have an "intensive" nuance, or as some put it, a causative function with a result emphasis (as contrasted with the Hiphil which shows causation with an agency emphasis; see Waltke & O'Connor, Introduction to Biblical Hebrew Syntax [Eisenbrauns, 1999] if this is of interest).
When this line of thinking is brought to bear on the case of yəḥôlēl in Ps 29:9, it should be apparent that an analysis along these lines can bear the "cause to give birth" nuance which the NET note prohibits.