Would this require a professional's help?
Although you could get a pro to help with this I believe the project is well within the grasp of a first-timer given the limited scope of what's needed here.
It is actually the preparatory stages are where most of the hands-on time and effort will be expended, by comparison the finishing steps (which you might possibly be more concerned with) require far less time, skill and effort to complete to a high standard.... finishing can be at core a simple process, it is only woodworkers who tend to unnecessarily complicate it :-)
Prep
From what we can see from the photos your board is flat enough (certainly sufficiently flat for a shelf) that it just needs to be smoothed off before finish application.
You can do this final smoothing by sanding or scraping. While I would normally recommend scraping over sanding there's a learning curve to scraping and a few gotchas, so if you're going to do all the work yourself sanding is probably your best option.
Sanding
Finish sanding is a process of progressively removing the scratches from the previous grit while keeping the surface flat.
I'm guessing the existing texture was produced by 80 grit, so starting at 120 is appropriate and you'll go from there. You don't need to sand to a particularly high grit so the progression of sanding grits you can use is fairly short, e.g. 120, 150, 220.
You can sand more finely than this (and many do, especially when power sanding) but beyond a certain point you see zero benefit after finish application.
For just the one shelf you can do this work entirely by hand; back the paper1 with a hard block and be careful not to 'lean over' when sanding near edges to maintain the flat surface and prevent unintentional rounding of crisp edges.
Sanding manually is a lot more work than with a random-orbit or other finish sander but for just one board it won't be too bad, and for a single project buying a power sander is arguably overkill. If you intend to do some future projects starting from bare wood however a sander does make more sense; although in that case I would strongly recommend you look into scraping as an option or at least an adjunct to sanding.
Finishing
Given your requirements for what you want it to look like you can use many common (traditional) finishing options as well as some modern products. I'm going to recommend three, in my approximate order of preference (please note that other people would order this list differently).
Wiping varnish
Everything you need to know to get started on this is in a previous Answer, here in an excerpt from Flexner on Finishing by Bob Flexner. In recent years wiping varnish has, understandably, become a very popular finishing option amongst amateurs and pros alike because it's nearly impossible to screw up. As a result there is tons more guidance available on this online if you feel you need to read or see more to get comfortable with the process before you try it for yourself. But I promise this is literally as easy as it seems.
Wiping varnish has one chief advantage over competitive wipe-on finishes such as "Scandinavian oil" and "Danish oil" and "Tung Oil Finish" (which are all basically the same3) in that it will dry hard through and through. It can be applied in exactly the same manner while providing more protection, although admittedly you're unlikely to need the extra protection for a shelf.
As you're finishing just one board for now and you'll be diluting at least 1:1, possibly as much as 1:3, only a small can will be needed. Don't forget to buy mineral spirits as well :-)
Lacquer
There's a good chance that your Herman Miller piece is finished in lacquer as it was the primary commercial finish through much of the second half of the 20th century — almost all MCM furniture has lacquer on it if still bearing their original finish.
This doesn't mean you need to use lacquer to match the look and feel however.
Many finishes are approximately equal in looks once fully dried, in fact the blended finishes mentioned above were originally formulated to match the lacquer finish on imported Scandi furniture!
Because you don't have spray equipment you're looking at rattle-can versions. This is in general an expensive way to buy any finish, but for a one-off it makes sense. In the US there are many brands (Deft, Mohawk, Rust-Oleum, ReRanch, Minwax to name a few) and different types on the market, so you'll need to do some shopping around if you want to go with lacquer. As long as you get an untinted or clear lacquer it doesn't matter hugely what type you get as far as final looks go, but an acrylic lacquer should be lower or low in VOCs making it a bit more user-friendly than some others.
Spraying a uniform coat is not rocket science but can take a couple of tries to get the knack. I suggest you use the underside to practice on. Don't fret if you get bad results, lacquers can be wiped off when fresh using solvent, or sanded off after drying if necessary.
A modern wax/oil finish
I wouldn't normally plug just one product but I'm going to specifically refer to Rubio Monocoat here. I don't use it myself and have no affiliation, this is based on the opinions of a few woodworkers I trust.
Monocoat seems to consistently perform, to live up to its hype.... in strong contrast to one or two competitor products.
It is incredibly easy to use, honestly even easier than wiping varnish. As you might expect the name refers to the fact that the manufacturer instructions are to apply just the one coat, but, numerous users report that they get a better-looking and more consistent finish by applying a second coat. So one or two coats and done, with an easy-as-pie application method.
So where's the downside? Depending on what you're used to paying for finish you might experience, ah, 'some' sticker shock — RRP for a smallish can is 70 bucks, versus $30-60 for nearly three times as much polyurethane4 O_O
Additional point:
I treat it with Feed-N-Wax Wood Polish & Conditioner about once a month.
You shouldn't need to polish a wooden surface anything like this often except for high-traffic items that see a lot of abrasion, or if using a product that wears off too easily.
With a hard finish in good condition you might only wax once a year, if that. Regular dusting and the occasional wipe with a damp cloth if needed is all that the bulk of furniture needs, with very occasional waxing for a modicum of added protection and for maintaining sheen (or adding back sheen lost to wear and cleaning).
A simple paste wax like the venerable Johnson's product is generally to be preferred for this purpose.
1 Paper is a shorthand here, "sandpaper" now commonly means actual sandpaper as well as film-backed, cloth-backed and screens-based abrasives.
2 Also bear in mind the top, front edge and maybe both ends need to be sanded more smoothly. Depending on the final positioning of the shelf the bottom might only be sanded further as practice for sanding the top.
3 They are some blend of varnish and oil, thinned down with additional spirits.
4 Which if converted to wiping varnish will be extended to double to quadruple the volume!
