Short Answer: The two accounts can be matched up with little difficulty. The focus of Genesis 2 merely expands on "Day 6" of creation. (This just leaves the question of Genesis 2:5, which is clearly just a commentary on the state of the earth at some point prior to the creation of man. So if the shrubs of 2:5 are included in the food of 1:11 then 2:5 is describing the state prior to 1:11. If they are different, then this is just a strategic point of reference for Moses' post-Fall audience.)
Let's start by examining the two sections carefully.
Genesis 1
Here is the chronology from Genesis 1:
Day 1: God creates light, and separates it from darkness
Day 2: God creates heaven to separate the lower waters from the upper waters
Day 3: God gathers the lower waters together, exposing dry land, and creates seed-bearing and fruit-bearing vegetation (for food for all the creatures of the earth)
Day 4: God creates the luminaries in the heavens to serve as lights, clocks, and calendars
Day 5: God creates the fish and birds
Day 6: God creates the land animals and man
Day 7: God rested
Genesis 1 definitely reads like a chronological account of the creation of the heavens and the earth. That is reason enough to consider the possibility that this is what it was intended to be. The reason it is not typically taken as such is that many interpreters know that it is "true" in some way, and yet also "know" that it could not possibly be literal, historical, or chronological. (Or so they were taught.)
Genesis 2
Here is the chronology from Genesis 2:
No shrub of the field was in the earth, and no plant of the field had sprouted because there was no rain, nor were there people to cultivate the ground
God formed Adam
God planted the Garden of Eden for Adam and stocked it with every tree that was good and pleasant
God placed Adam in the Garden
God decided to make Adam a helper, and started by bringing every kind of bird and beast out of the ground and to Adam so he could name them
God made Eve
The focus of this chapter is clearly on the creation of Adam and Eve and the Garden paradise that God made for them to dwell in. The previous chapter covered the creation of the entire universe. The next chapter covers Adam and Eve falling into sin, the cursing of the ground, and their exile from the Garden. If we follow the flow of the narrative through the first 3 chapters of Genesis it is easy to see that Genesis 2 expands on Genesis 1 and prepares the reader for Genesis 3.
Can They Be Merged?
As mentioned, Genesis 1 is presented to the reader as a chronological account of the creation of the universe (with a heavy focus on man and man's perspective.) Genesis 2 is presented to the reader as an account of the creation of Adam and Eve and the original Garden paradise. Genesis 3 is presented to the reader as a grand "mucking up" of everything good that God made.
With that in mind, we can start by recognizing a few things:
The main focus of Genesis 2 is the creation of Adam and Eve. This took place during "Day 6" of the Genesis 1 account.
The purpose of Genesis 2 is to prepare the reader to understand Genesis 3, not to provide stale historical records for future generations to fill their heads with. Everything mentioned in Genesis 2 is strategic, with an eye toward Genesis 3 and beyond.
The audience of Genesis 1-3 lived after the Fall and did not experience life during the Creation Week, or life in the Garden paradise.
So we can attempt an initial chronology based on these two accounts:
God made light, heaven, and dry land
God made seed-bearing and fruit-bearing vegetation for food
God made fish, birds, and land animals
God made Adam
God made the Garden of Eden and placed Adam there
God had Adam name the animals and then He made Eve
So the two chronologies mesh perfectly. At this point the only real question is what Genesis 2:5 means:
Now no shrub of the field was yet in the earth, and no plant of the field had yet sprouted, for the Lord God had not sent rain upon the earth, and there was no man to cultivate the ground.
The question is how does this affect our chronology? Actually, the answer is pretty simple: It doesn't! We already know that the author is going back and elaborating on stuff that had already been described in the previous chapter, so all that this verse is saying is that at some point prior to the creation of man the "shrubs of the field" and "plants of the field" had not yet sprouted. And actually, we already knew that from Genesis 1.
Supplementary Discussion
Of course, this leaves us with an obvious question: Why did Moses include Genesis 2:5 at the beginning of his account of the creation of Adam and Eve and the Garden of Eden.
The easy answer is that he is presenting the earth as a place void of man, and then proceeds to describe the creation of man. But there may in fact be an additional reason. Genesis 3 touches on trees, Adam, Eve, cursed ground, and exile from Eden. Genesis 2 touches on trees, Adam, Eve, and Eden. Verse 5 may in fact touch on the "cursed ground."
After the Fall, the ground was cursed, and it was by thorns and hard work that the earth would yield its fruit. Before sin, God placed man in a garden where every pleasant and good tree was provided to them by God for food. Moses wants us to see the contrast there. Verse 5 may simply be a reference for his post-Fall audience to recall that the land at that time was not yet full of the thorns and "shrubs of the field" and "plants of the field" which required rain and cultivation before they could sprout.
So then, the point of Genesis 2:5 is not to claim that in Day 6 of the creation week there was not yet any vegetation on the earth (Moses is not an idiot!)... but rather, he is explaining to his readers that originally, before sin (and before man was on the earth,) the effects of the curse were not yet in the earth, and God provided for His creation perfectly.